MATH 251 Lecture 9

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Differential of a Function

(Section 12.4)

The derivative of a function at a point $p$ is the best linear approximation of $f(\overrightarrow{x})-f(p)$

$\begin{array}{rl}\mathrm{d}f& =\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial y}\mathrm{d}y\\ & =f_x\mathrm{d}x+f_y\mathrm{d}y\\ & =\overrightarrow{\mathrm{\nabla }}f\cdot ⟨\mathrm{d}x,\mathrm{d}y⟩\\ & =\overrightarrow{\mathrm{\nabla }}f\cdot \mathrm{d}\overrightarrow{s}\end{array}$

Example

f(x) is a single variable function.

$f(x+h)=f(x)+f^{\prime }(x)\cdot h+O(h^2)$, where $f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

The best approximation for $f(x+h)-f(x)$ is $f^{\prime }(x)\cdot h$

Multivariable

For $p=(x_0,y_0)$

$\begin{array}{rl}f(x_0+h,y_0+k)-f(x_0,y_0)& \approx \frac{\partial f}{\partial x}(x_0,y_0)\cdot h+\frac{\partial f}{\partial y}(x_0,y_0)\cdot k\\ & \approx f_x(x_0,y_0)h+f_y(x_0,y_0)k\end{array}$

Example

Estimate $f(1.1,1.2)$, where $f(x,y)={\mathrm{e}}^{xy}\sin (x^2-y^2)$

$\mathrm{d}f=\{\begin{array}{l}{f}_x=y{\mathrm{e}}^{xy}\sin (x^2-y^2)+2x{\mathrm{e}}^{xy}\cos (x^2-y^2)\\ f_y=x{\mathrm{e}}^{xy}\sin (x^2-y^2)-2y{\mathrm{e}}^{xy}\cos (x^2-y^2)\end{array}$

$\begin{array}{rl}f(1,1)& =0\\ f(1.1,1.2)& \approx -0.2\mathrm{e}\\ \mathrm{d}x=.1,\mathrm{d}y=.2\\ \mathrm{d}f& =f(1,1)+f(1.1,1.2)\approx -.2\mathrm{e}\end{array}$

Tangent Plane

$z-z_0=f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)$