MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 28 July 2014

Lecture 3

A function takes in a number, chews on it and spits out another number $$\text{input number}\hspace{0.17em}x⟶\begin{array}{c} f \end{array}⟶\text{output number}\hspace{0.17em}f\left(x\right)\text{.}$$ A constant function always spits out the same number, no matter what the input is.

$f\left(x\right)=2\text{.}$ $$x⟶\begin{array}{c} f \end{array}⟶2$$ If $x=3\pi +7$ then $f\left(x\right)=2\text{.}$
If $x=3+3i$ then $f\left(x\right)=2\text{.}$
If $x=1$ then $f\left(x\right)=2\text{.}$
If $x=\sqrt{7}$ then $f\left(x\right)=2\text{.}$

We call this function $2\text{.}$ $$5⟶\begin{array}{c} 2 \end{array}⟶2\sqrt{10}⟶\begin{array}{c} 2 \end{array}⟶2$$ So $2$ sometimes means the number $2$ and sometimes means the function $2\text{.}$

Derivatives

A derivative takes in a function, chews on it and spits out another function $$f⟶\begin{array}{c} ddx \end{array}⟶\frac{df}{dx}\text{.}$$ The derivative $\frac{d}{dx}$ spits out a function according to the rules:

(1)	$\frac{dx}{dx}=1\text{.}$
(2)	$\frac{d\left(cf\right)}{dx}=c\frac{df}{dx},$ if $c$ does not change when $x$ changes.
(3)	$\frac{d(f+g)}{dx}=\frac{df}{dx}+\frac{dg}{dx}\text{.}$
(4)	$\frac{d\left(fg\right)}{dx}=f\frac{dg}{dx}+\frac{df}{dx}g\text{.}$

Find $\frac{dy}{dx}$ if $y=5x\text{.}$ $$\frac{dy}{dx}=\frac{d\left(5x\right)}{dx}=5\frac{dx}{dx}=5·1=5\text{.}$$

Find $\frac{dy}{dx}$ if $y=\pi x\text{.}$ $$\frac{dy}{dx}=\frac{d\left(\pi x\right)}{dx}=\pi \frac{dx}{dx}=\pi ·1=\pi \text{.}$$

Find $\frac{dy}{dx}$ if $y=1\text{.}$ $$\frac{dy}{dx}=\frac{d1}{dx}=\frac{d(1·1)}{dx}=1·\frac{d1}{dx}+\frac{d1}{dx}·1=\frac{d1}{dx}+\frac{d1}{dx}\text{.}$$ Subtract $\frac{d1}{dx}$ from both sides. So $$\frac{d1}{dx}=0\text{.}$$

Find $\frac{dy}{dx}$ if $y=5\text{.}$ $$\frac{dy}{dx}=\frac{d5}{dx}=\frac{d(5·1)}{dx}=5\frac{d1}{dx}=5·0=0\text{.}$$

Find $\frac{dy}{dx}$ if $y=6342\text{.}$ $$\frac{d6342}{dx}=\frac{d(6342·1)}{dx}=6342\frac{d1}{dx}=6342·0=0\text{.}$$

Find $\frac{dc}{dx}$ if $c$ is a constant. $$\frac{dc}{dx}=\frac{d(c·1)}{dx}=c\frac{d1}{dx}=c·0=0\text{.}$$

Find $\frac{dy}{dx}$ if $y=3x+12\text{.}$ $$\frac{dy}{dx}=\frac{d(3x+12)}{dx}=\frac{d\left(3x\right)}{dx}+\frac{d12}{dx}=3\frac{dx}{dx}+0=3·1+0=3\text{.}$$

Find $\frac{dy}{dx}$ if $y=x^2\text{.}$ $$\frac{dy}{dx}=\frac{d{x}^2}{dx}=\frac{dx·x}{dx}=x·\frac{dx}{dx}+=x·1+1·x=2x\text{.}\frac{dx}{dx}·x$$

Find $\frac{d{x}^3}{dx}\text{.}$ $$\frac{d{x}^3}{dx}=\frac{d(x^2·x)}{dx}=x^2\frac{dx}{dx}+\frac{d{x}^2}{dx}·x=x^2·1+2x·x=3x^2\text{.}$$

Find $\frac{d{x}^4}{dx}\text{.}$ $$\frac{d{x}^4}{dx}=\frac{d(x^3·x)}{dx}=x^3·\frac{dx}{dx}+\frac{d{x}^3}{dx}·x=x^3·1+3x^2·x=4x^3\text{.}$$

$$⋮$$

Find $\frac{d{x}^{6342}}{dx}\text{.}$ dx6342dx= d(x6341·x)dx= x6341·dxdx+ dx6341dx·x= x6341·1+6341x6340·x= 6342x6341.

Find $\frac{d{x}^n}{dx}$ if $n=1,2,3,\dots \text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{d{x}^n}{dx}}& =& {\displaystyle \frac{d(x^{n-1}·x)}{dx}}\\ & =& {\displaystyle x^{n-1}·\frac{dx}{dx}+\frac{d{n}^{n-1}}{dx}·x}\\ & =& {\displaystyle x^{n-1}·1+(n-1)x^{n-2}·x\left(\genfrac{}{}{0ex}{}{\text{since we just found}}{\frac{d{x}^{n-1}}{dx}=(n-1)x^{n-2}}\right)}\\ & =& {\displaystyle x^{n-1}+(n-1)x^{n-1}\text{.}}\end{array}$$ So $$\frac{d{x}^n}{dx}=n{x}^{n-1}\text{.}$$

Find $\frac{d{x}^n}{dx}$ if $n=0\text{.}$ $$\frac{d{x}^n}{dx}=\frac{d{x}^0}{dx}=\frac{d1}{dx}=0=n{x}^{n-1}\text{.}$$

Find $\frac{d{x}^{-6342}}{dx}\text{.}$ $$\frac{d(x^{-6342}·x^{6342})}{dx}=\frac{d{x}^0}{dx}=\frac{d1}{dx}=0\text{.}$$ On the other hand, $$\begin{array}{ccc}{\displaystyle \frac{d(x^{-6342}·x^{6342})}{dx}}& =& {\displaystyle x^{-6342}\frac{d{x}^{6342}}{dx}+\frac{d{x}^{-6342}}{dx}·x^{6342}}\\ & =& {\displaystyle x^{-6342}6342x^{6341}+\frac{d-6342}{dx}·x^{6342}\text{.}}\end{array}$$ So $$0=6342x^{-1}+x^{6342}\frac{d{x}^{-6342}}{dx}\text{.}$$ So $$\begin{array}{ccc}{\displaystyle \frac{d{x}^{-6342}}{dx}}& =& {\displaystyle -6342x^{-1}{x}^{-6342}}\\ & =& {\displaystyle (-6342)x^{-6342}\text{.}}\end{array}$$

Find $\frac{d{x}^{-n}}{dx}$ if $n=1,2,3,\dots \text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{d{x}^n{x}^{-n}}{dx}}& =& {\displaystyle x^n\frac{d{x}^{-n}}{dx}+\frac{d{x}^n}{dx}{x}^{-n}}\\ & =& {\displaystyle x^n\frac{d{x}^{-n}}{dx}+n{x}^{n-1}{x}^{-n}}\\ & =& {\displaystyle x^n\frac{d{x}^{-n}}{dx}+n{x}^{-1}\text{.}}\end{array}$$ On the other hand $$\frac{d\left(x^n{x}^{-n}\right)}{dx}=\frac{d{x}^0}{dx}=\frac{d1}{dx}=0\text{.}$$ So $$x^n\frac{d{x}^{-n}}{dx}+n{z}^{-1}=0\text{.}$$ Solve for $\frac{d{x}^{-n}}{dx}\text{.}$ Then $$\frac{d{x}^{-n}}{dx}=-n{x}^{-1}{x}^{-n}=-n{x}^{-n-1}\text{.}$$

Let $y=3x^3+5x^2+2x+7\text{.}$ Find $\frac{dy}{dx}\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}}& =& {\displaystyle \frac{d(3x^3+5x^{}+2\mathrm{x+<mn>7</mn>><mo>)</mo>}}{dx}}\\ & =& {\displaystyle \frac{d\left(3x^3\right)}{dx}+\frac{d(5x^2+2x+7)}{dx}}\\ & =& {\displaystyle \frac{d\left(3x^3\right)}{dx}+\frac{d\left(5x^2\right)}{dx}+\frac{d\left(2x\right)}{dx}+\frac{d7}{dx}}\\ & =& {\displaystyle 3\frac{d{x}^3}{dx}+5\frac{d{x}^2}{dx}+2\frac{dx}{dx}+7\frac{d1}{dx}}\\ & =& {\displaystyle 3·3x^2+5·2x+2·1+7·0}\\ & =& {\displaystyle 9x^2+10x+2\text{.}}\end{array}$$

If $y=-7x^{-13}+5x^{-7}+(6+2i)x^{38}\text{.}$ Find $\frac{dy}{dx}\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}}& =& {\displaystyle \frac{d(-7x^{-13}+5x^{-7}+(6+2i)x^{38})}{dx}}\\ & =& {\displaystyle \frac{d(-7x^{-13})}{dx}+\frac{d\left(5x^{-7}\right)}{dx}+\frac{d\left((6+2i)x^{38}\right)}{dx}}\\ & =& {\displaystyle -7\frac{d{x}^{-13}}{dx}+5\frac{d{x}^{-7}}{dx}+(6+2i)\frac{d{x}^{38}}{dx}}\\ & =& {\displaystyle (-7)(-13)x^{-13-1}+5(-7)x^{-7-1}+(6+2i)38x^{38-1}}\\ & =& {\displaystyle 91x^{-14}-35x^{-8}+(228+76i)x^{37}\text{.}}\end{array}$$

Notes and References

These are a typed copy of Lecture 3 from a series of handwritten lecture notes for the class MATH 221 given on September 11, 2000.

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