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 $f({\vec {x}})-f(p)$

${\begin{aligned}\mathrm {d} f&={\frac {\partial f}{\partial x}}\,\mathrm {d} x+{\frac {\partial f}{\partial y}}\,\mathrm {d} y\\&=f_{x}\,\mathrm {d} x+f_{y}\,\mathrm {d} y\\&={\vec {\nabla }}f\cdot \left\langle \mathrm {d} x,\mathrm {d} y\right\rangle \\&={\vec {\nabla }}f\cdot \mathrm {d} {\vec {s}}\end{aligned}}$

Example

f(x) is a single variable function.

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

The best approximation for $f(x+h)-f(x)$ is $f'(x)\cdot h$

Multivariable

For $p=\left(x_{0},\,y_{0}\right)$

${\begin{aligned}f\left(x_{0}+h,y_{0}+k\right)-f\left(x_{0},y_{0}\right)&\approx {\frac {\partial f}{\partial x}}\left(x_{0},y_{0}\right)\cdot h+{\frac {\partial f}{\partial y}}\left(x_{0},y_{0}\right)\cdot k\\&\approx f_{x}\left(x_{0},y_{0}\right)\,h+f_{y}(x_{0},y_{0})\,k\end{aligned}}$

Example

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

$\mathrm {d} f={\begin{cases}f_{x}=y\mathrm {e} ^{xy}\sin {\left(x^{2}-y^{2}\right)}+2x\mathrm {e} ^{xy}\cos {\left(x^{2}-y^{2}\right)}\\f_{y}=x\mathrm {e} ^{xy}\sin {\left(x^{2}-y^{2}\right)}-2y\mathrm {e} ^{xy}\cos {\left(x^{2}-y^{2}\right)}\end{cases}}$

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

Tangent Plane

$z-z_{0}=f_{x}(x_{0},y_{0})\cdot (x-x_{0})+f_{y}(x_{0},y_{0})\cdot (y-y_{0})$