MATH 409 Lecture 15

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Lecture Slides

Review

Derivative is defined as $f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$ or equivalently $f'(a)=\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}$.

Notations:

• Lagrange: $f'$
• Newton: ${\dot {f}}$
• Leibniz: ${\frac {\mathrm {d} f}{\mathrm {d} x}}$
• Euler: $D_{x}\,f$
• $f^{(1)}$ (higher order derivatives)

Derivative at a point denoted $\left.(f(x))'\right|_{x=a}$.

Differentiability Theorems

• Sum and Difference Rules
• Product Rule
• Quotient Rule

Derivatives of Elementary Functions

Higher-Order Derivatives

Defined inductively:

For $n\geq 2$ for any $a\in \mathbb {R}$, the $n$-th derivative of $f$ at a point $a$, denoted $f^{(n)}(a)$, is defined by $f^{(n)}(a)=(f^{(n-1)})'(a)$.

Derivative Spaces

Let $I$ be an interval of the real line $\mathbb {R}$. We denote $C(I)$ or $C^{0}(I)$ as the set of all continuous functions on $I$.

For any $n\in \mathbb {N}$, we denote $C^{n}(I)$ as the set of all functions that are $n$ times continuously differentiable on $I$. (that is, the $n$-th derivative is continuous)

$C^{\infty }(I)$ represents the set of all infinitely continuously differentiable on $I$.

Examples

$f(0)=0$, and $f(x)=x\sin {\frac {1}{x}}$ for $x\neq 0$.

Product and Chain rules show that $f$ is differentiable on $\mathbb {R} \setminus \left\{0\right\}$.

For $x\neq 0$.

$f'(x)=\left(x\,\sin {\frac {1}{x}}\right)'=\sin {\frac {1}{x}}+x\,\left(\sin {\frac {1}{x}}\right)=\sin {\frac {1}{x}}-{\frac {1}{x}}\,\cos {\frac {1}{x}}$

The function $f$ is continuous at $0$, but the derivative is not continuous at $0$, so $f$ is not differentiable at $0$. Indeed ${\frac {f(h)-f(0)}{h}}=\sin {\frac {1}{h}}$ has no limit as $h\to 0$.

$g(0)=0$, and $g(x)=x^{2}\sin {\frac {1}{x}}$ for $x\neq 0$.

As above, the Product and Chain rules show that $f$ is differentiable on $\mathbb {R} \setminus \left\{0\right\}$.

For $x\neq 0$,

${\begin{aligned}g'(x)&=\left(x\cdot x\sin {\frac {1}{x}}\right)'\\&=2x\,\sin {\frac {1}{x}}-\cos {\frac {1}{x}}\end{aligned}}$

The function is differentiable at 0. Indeed ${\frac {g(h)-g(0)}{h}}=h\,\sin {\frac {1}{h}}\to 0$ as $h\to 0$.

Note that $g$ is not continuously differentiable on $\mathbb {R}$ since $g'$ is not continuous at $0$. Namely, $\lim _{x\to 0}g'(x)$ does not exist.

Power Rule

Theorem. $(x^{n})'=n\,x^{n-1}$ for all $x\in \mathbb {R}$ and $n\in \mathbb {N}$.

Proof by induction. In the case $n=1$, we have $x'=1=1\,x^{0}$ for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} .

Assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x^n)' = n \, x^{n-1}} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} and all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} . Using the product rule, we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x^{n+1} \right)' = \left( x^n \, x \right)' = (x^n)' \, x + x^n \, x' = n \, x^{n-1} \, x + x^n = (n+1) \, x^n} .

quod erat demonstrandum

Note: The theorem can also be proved using the formula ${\frac {x^{n}-a^{n}}{x-a}}=x^{n-1}+x^{n-2}\,a+\dots +x\,a^{n-2}+a^{n-1}$.

In the same breath, $(x^{-n})'=-n\,x^{-n-1}$ for all $x\neq 0$ and $n\in \mathbb {N}$.

Using reciprocal rule, we obtain $(x^{-n})'=\left({\frac {1}{x^{n}}}\right)'={\frac {-(x^{n})'}{(x^{n})^{2}}}={\frac {-n\,x^{n-1}}{x^{2n}}}=-n\,x^{-n-1}$

Derivative of Inverse Function

Theorem. Suppose $f$ is an invertible continuous function. If $f$ is differentiable at a point $a$ and $f'(a)\neq 0$, then the inverse function is differentiable at the point $b=f(a)$ and

$(f^{-1})'(b)={\frac {1}{f'(a)}}={\frac {1}{f'(f^{-1}(b))}}$

Proof. Since $f$ is differentiable at $a$, we know that $f$ is defined on an open interval $I=(c,d)$ containing $a$. Since $f$ is continuous and invertible, it follows from the Intermediate Value Theorem that $f$ is strictly monotone on $I$, the image $f(I)$ is an open interval containing $b$, and the inverse function $f^{-1}$ is continuous and strictly monotone on $f(I)$.

We have $\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}=f'(a)$. Since $f'(a)\neq 0$, it follows that $\lim _{x\to a}{\frac {x-a}{f(x)-f(a)}}={\frac {1}{f'(a)}}$. Since $f^{-1}$ is continuous and monotone on the interval $f(I)$, we obtain that $f^{-1}(y)\to a$ and $f^{-1}(y)\neq a$ when $y\to b$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \ne b} .

Therefore $\lim _{y\to b}{\frac {f^{-1}(y)-a}{y-b}}=\lim _{y\to b}{\frac {f^{-1}(y)-a}{f(f^{-1}(y))-b}}=\lim _{x\to a}{\frac {x-a}{f(x)-f(a)}}={\frac {1}{f'(a)}}$.

quod erat demonstrandum

Remark. In the case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(a) = 0} , the inverse function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} is not differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)} .

Indeed, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} is differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = f(a)} , the chain rule implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( f^{-1} \circ f \right)'(a) = (f^{-1})'(b) \cdot f'(a)} . Obviously the LHS is the identity function, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f^{-1} \circ f)'(a) = 1 \ne 0} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(a) \ne 0} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \arccos{x}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in [-1, 1]} .

The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y) = \cos{y}} is strictly decreasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0,\pi \right]} and maps this interval onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -1, 1 \right]} . By definition, the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \arccos{x}} is the inverse of the restriction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0, \pi \right]} . Notice that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g'(0) = g'(\pi) = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g'(y) \ne 0} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in \left( 0, \pi \right)} . It follows that the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( -1, 1 )} and not differentiable at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} . Moreover, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \left( -1,1 \right)} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \frac{1}{g'(f(x))} = -\frac{1}{\sin{\arccos{x}}}}

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \arccos{x}} (hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \cos{y}} ). We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2{y} = \cos^2{y} = 1} by the pythagorean identity. Besides, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{y} > 0} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in (0, \pi)} . Consequently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{y} = \sqrt{1-\cos^2{y}} = \sqrt{1-x^2}} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = -\frac{1}{\sqrt{1-x^2}}} .

quod erat demonstrandum

Homework hint; use similar method to prove for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsin{x}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan{x}} , just use same identity (divide by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2{y}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan{x}} )

Exponential and Logarithmic Functions

Theorem. The sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n = \left( 1 + \frac{1}{n} \right)^n} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} is increasing and bounded, hence convergent.

The limit is the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e} = 2.718281828 \ldots } (number of letters in each word in "I'm forming a mnemonic to remember a constant in analysis")

Corollary. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 0} \left( 1 + x \right)^{\frac{1}{x}} = \mathrm{e}} .

Not proved here... (Ain't nobody got time for that!)

for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ne 1} , the exponential function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a^x} is strictly monotone and continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} . It maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( 0, \infty \right)} . Therefore the inverse function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y) = \log_a{y}} is strictly monotone and continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( 0, \infty \right)} . The natural logarithm $\log _{e}{y}$ is also denoted just $\log {y}$.

Since $\left(1+h\right)^{\frac {1}{h}}\to \mathrm {e}$ as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} , it follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^{-1} \log{\left( 1+h \right)} = \log{\left( 1 + h \right)^{\frac{1}{h}}} \to \log{\mathrm{e}} = 1} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \to 0} . In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \left( \log{y} \right)' \right|_{y=1} = 1}

Examples

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \mathrm{e}^x} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x+h) - f(x)}{h} = \frac{e^x(e^h-1)}{h}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,h \in \mathbb{R}} . Therefore for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \lim_{h \to 0} \frac{e^h-1}{h} = e^x \, f'(0) = e^x}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a^x} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}} , where $a>0$.

Equivalently, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \mathrm{e}^{\log{a^x}} = \mathrm{e}^{x \, \log{a}}} . So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \mathrm{e}^{x \, \log{a}} \, \log{a} = a^x \, \log{a}} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \log{x}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \left( 0, \infty \right)} .

Since $f$ is the inverse function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y) = \mathrm{e}^{y}} , we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \frac{1}{g'(\log{x})} = \frac{1}{\mathrm{e}^{\log{x}}} = \frac{1}{x}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 0} .

Power Rule: General Case

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x^\alpha \right)' = \alpha \, x^{\alpha-1}} for all $x>0$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in \mathbb{R}} .

Proof. Let us fix a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in \mathbb{R}} and consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^\alpha} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \left( 0, \infty \right)} . For any $x>0$, we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \mathrm{e}^{\log{x^\alpha}} = a^{\log{x}}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = \mathrm{e}^{\alpha}} . Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = h \circ g} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = \log{x}} for $x>0$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(y) = a^y} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in \mathbb{R}} . By the chain rule, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \alpha \, x^{\alpha-1}} .

quod erat demonstrandum