MATH 221

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

Last updated: 30 July 2014

Lecture 4

Last time we pointed out that there are different kinds of derivatives: $$\begin{array}{ccc}\text{Derivative with respect to}\hspace{0.17em}x& & \text{Derivative with respect to}\hspace{0.17em}g\\ f⟶\begin{array}{c} ddx \end{array}⟶\frac{df}{dx}& & f⟶\begin{array}{c} ddg \end{array}⟶\frac{df}{dg}\\ \text{This one satisfies}& & \text{This one satisfies}\\ \begin{array}{ccc}{\displaystyle \frac{dx}{dx}}& =& {\displaystyle 1}\\ {\displaystyle \frac{d(y+z)}{dx}}& =& {\displaystyle \frac{dy}{dx}+\frac{dz}{dx}}\\ {\displaystyle \frac{d\left(cy\right)}{dx}}& =& {\displaystyle c\frac{dy}{dx},\text{if}\hspace{0.17em}c\hspace{0.17em}\text{is a constant}}\\ {\displaystyle \frac{d\left(yz\right)}{dx}}& =& {\displaystyle y\frac{dz}{dx}+\frac{dy}{dx}z}\end{array}& & \begin{array}{ccc}{\displaystyle \frac{dg}{dg}}& =& {\displaystyle 1}\\ {\displaystyle \frac{d(y+z)}{dg}}& =& {\displaystyle \frac{dy}{dg}+\frac{dz}{dg}}\\ {\displaystyle \frac{d\left(cy\right)}{dg}}& =& {\displaystyle c\frac{dy}{dg},\text{if}\hspace{0.17em}c\hspace{0.17em}\text{is a constant}}\\ {\displaystyle \frac{d\left(yz\right)}{dz}}& =& {\displaystyle y\frac{dz}{dg}+\frac{dy}{dg}z}\end{array}\end{array}$$ What is the relation between $\frac{df}{dx}$ and $\frac{df}{dg}\text{??}$ $$\begin{array}{cc}⟶\begin{array}{c} ddg \end{array}⟶& ⟶\begin{array}{c} ddx \end{array}⟶\\ \begin{array}{ccc}{\displaystyle \frac{d{g}^0}{dg}}& =& {\displaystyle \frac{d1}{dg}=0}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d{g}^0}{dx}}& =& {\displaystyle \frac{d1}{dx}=0}\end{array}\\ \begin{array}{ccc}{\displaystyle \frac{dg}{dg}}& =& {\displaystyle 1}\end{array}& \begin{array}{ccc}{\displaystyle \frac{dg}{dx}}& =& {\displaystyle \frac{dg}{dx}}\end{array}\\ \begin{array}{ccc}{\displaystyle \frac{d{g}^2}{dg}}& =& {\displaystyle \frac{d(g·g)}{dg}}\\ & =& {\displaystyle g\frac{dg}{dg}+\frac{dg}{dg}g}\\ & =& {\displaystyle g+g=2g}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d{g}^2}{dx}}& =& {\displaystyle \frac{d(g·g)}{dx}}\\ & =& {\displaystyle g\frac{dg}{dx}+\frac{dg}{dx}·g}\\ & =& {\displaystyle 2g\frac{dg}{dx}}\end{array}\\ \begin{array}{ccc}{\displaystyle \frac{d{g}^3}{dg}}& =& {\displaystyle \frac{d(g^2·g)}{dg}}\\ & =& {\displaystyle g^2\frac{dg}{dg}+\frac{d{g}^2}{dg}g}\\ & =& {\displaystyle g^2+2g·g=3g^2}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d{g}^3}{dx}}& =& {\displaystyle \frac{d(g^2·g)}{dx}}\\ & =& {\displaystyle g^2\frac{dg}{dx}+\frac{d{g}^2}{dx}g}\\ & =& {\displaystyle g^2\frac{dg}{dx}+2g\frac{dg}{dx}g}\\ & =& {\displaystyle g^2\frac{dg}{dx}+2g^2\frac{dg}{dx}}\end{array}\\ \begin{array}{ccc}{\displaystyle \frac{d{g}^4}{dg}}& =& {\displaystyle \frac{d(g^4·g)}{dg}}\\ & =& {\displaystyle g^3\frac{dg}{dg}+\frac{d{g}^3}{dg}·g}\\ & =& {\displaystyle g^3+3g^2·g}\\ & =& {\displaystyle 4g^3}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d{g}^4}{dx}}& =& {\displaystyle \frac{d(g^3·g)}{dx}}\\ & =& {\displaystyle g^3\frac{dg}{dx}+\frac{d{g}^3}{dx}g}\\ & =& {\displaystyle g^3\frac{dg}{dx}+3g^2\frac{dg}{dx}g}\\ & =& {\displaystyle 4g^3\frac{dg}{dx}}\end{array}\\ ⋮& ⋮\\ \begin{array}{ccc}{\displaystyle \frac{d{g}^{6342}}{dg}}& =& {\displaystyle 6342g^{6341}}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d{g}^{6342}}{dx}}& =& {\displaystyle 6342g^{6341}\frac{dg}{dx}}\end{array}\\ \begin{array}{ccc}{\displaystyle \frac{d(3g^2+2g+7)}{dg}}& =& {\displaystyle \frac{d\left(3g^2\right)}{dg}+\frac{d\left(2g\right)}{dg}+\frac{d7}{dg}}\\ & =& {\displaystyle 3\frac{d{g}^2}{dg}+2\frac{dg}{dg}+0}\\ & =& {\displaystyle 3·2g+2·1}\\ & =& {\displaystyle 6g+2}\end{array}& \begin{array}{ccc}{\displaystyle \frac{d(3g^2+2g+7)}{dx}}& =& {\displaystyle \frac{d\left(3g^2\right)}{dx}+\frac{d\left(2g\right)}{dx}+\frac{d7}{dx}}\\ & =& {\displaystyle 3\frac{d{g}^2}{dx}+2\frac{dg}{dx}+\frac{d7}{dx}}\\ & =& {\displaystyle 3·2g\frac{dg}{dx}+2\frac{dg}{dx}+0}\\ & =& {\displaystyle 6g\frac{dg}{dx}+2\frac{dg}{dx}}\\ & =& {\displaystyle (6g+2)\frac{dg}{dx}}\end{array}\end{array}$$ If $f$ is any function, then we have the chain rule $$\frac{df}{dx}=\frac{df}{dg}\frac{dg}{dx},$$ i.e. $$f⟶\begin{array}{c} ddx \end{array}⟶\frac{dg}{dx}\frac{dg}{dx}f⟶\begin{array}{c} ddg \end{array}⟶\frac{df}{dg}\text{.}$$

Find $\frac{dy}{dx}$ when $y={(2x-5)}^2\text{.}$

If $g=2x-5$ then $y=g^2\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}=\frac{dy}{dg}\frac{dg}{dx}}& =& {\displaystyle \frac{d{g}^2}{dg}\frac{dg}{dx}=2g\frac{d(2x-5)}{dx}=2g\left(2\right)}\\ & =& {\displaystyle 4(2x-5)\text{.}}\end{array}$$

Find $\frac{d}{dx}$ when $y={(3x-4)}^3\text{.}$

If $g=3x-4$ then $y=g^3\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}=\frac{dy}{dg}\frac{dg}{dx}}& =& {\displaystyle \frac{d{g}^3}{dg}\frac{dg}{dx}=3g^2\frac{dg}{dx}=3(3x-4)\frac{d(3x-4)}{dx}}\\ & =& {\displaystyle 3(3x-4)·3=27x-36\text{.}}\end{array}$$

Find $\frac{dy}{dx}$ when $y={(2x-5)}^2{(3x-4)}^3\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}}& =& {\displaystyle \frac{d{(2x-5)}^2{(3x-4)}^3}{dx}}\\ & =& {\displaystyle {(2x-5)}^2\frac{d{(3x-4)}^3}{dx}+\frac{d{(2x-5)}^2}{dx}{(3x-4)}^3}\\ & =& {\displaystyle {(2x-5)}^{2}3{(3x-4)}^2·\frac{d(3x-4)}{dx}+{(3x-4)}^{3}2(2x-5)\frac{d(2x-5)}{dx}}\\ & =& {\displaystyle 3{(2x-5)}^2{(3x-4)}^2·3+2(2x-5){(3x-4)}^3·2}\\ & =& {\displaystyle (2x-5){(3x-4)}^2(9(2x-5)+4(3x-4))}\\ & =& {\displaystyle (2x-5){(3x-4)}^2(30x-61)\text{.}}\end{array}$$

Find $\frac{dy}{dx}$ when $y={\left(\frac{x-3}{x-4}\right)}^2\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{d{\left(\frac{x-3}{x-4}\right)}^2}{dx}}& =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)\frac{d\left(\frac{x-3}{x-4}\right)}{dx}}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)\frac{d\left((x-3){(x-4)}^{-1}\right)}{dx}}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)((x-3)\frac{d{(x-4)}^{-1}}{dx}+\frac{d(x-3)}{dx}{(x-4)}^{-1})}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)((x-3)(-1){(x-4)}^{-2}\frac{d(x-4)}{dx}+1·{(x-4)}^{-1})}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)(-\frac{(x-3)}{{(x-4)}^2}·1+\frac{1}{x-4})}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)(-\frac{x+3}{{(x-4)}^2}+\frac{x-4}{{(x-4)}^2})}\\ & =& {\displaystyle 2\left(\frac{x-3}{x-4}\right)\frac{(-1)}{{(x-4)}^2}}\\ & =& {\displaystyle \frac{-2x+6}{{(x-4)}^3}}\end{array}$$

Find $\frac{d{x}^{\frac{m}{n}}}{dx}\text{.}$ $$\frac{d{\left(x^{\frac{m}{n}}\right)}^n}{dx}=\frac{d{x}^m}{dx}=m{x}^{m-1}\text{.}$$ On the other hand $$\frac{d{\left(x^{\frac{m}{n}}\right)}^n}{dx}=n{\left(x^{\frac{m}{n}}\right)}^{n-1}\frac{d{x}^{\frac{m}{n}}}{dx}\text{.}$$ So $$m{x}^{m-1}=n{\left(x\frac{m}{n}\right)}^{n-1}\frac{d{x}^{\frac{m}{n}}}{dx}$$ and we can solve for $\frac{d{x}^{\frac{m}{n}}}{dx}\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{d{x}^{\frac{m}{n}}}{dx}}& =& {\displaystyle \frac{m{x}^{m-1}}{n{\left(x^{\frac{m}{n}}\right)}^{n-1}}=\frac{m{x}^{m-1}}{n{\left(x^{\frac{m}{n}}\right)}^n{\left(x^{\frac{m}{n}}\right)}^{-1}}}\\ & =& {\displaystyle \frac{m{x}^{m-1}}{m{x}^m\frac{1}{x^{\frac{m}{n}}}}=\frac{m}{n}{x}^{-1}{x}^{\frac{m}{n}}=\frac{m}{n}{x}^{\frac{m}{n}=1}\text{.}}\end{array}$$ So $$\frac{d{x}^{\frac{m}{n}}}{dx}=\frac{m}{n}{x}^{\frac{m}{n}-1}\text{.}$$

Find $\frac{dy}{dx}$ when $y=\frac{x}{\sqrt{1-2x}}\text{.}$ $$\begin{array}{ccc}{\displaystyle \frac{dy}{dx}}& =& {\displaystyle \frac{d\frac{x}{\sqrt{1-2x}}}{dx}}\\ & =& {\displaystyle \frac{dx{\left(\sqrt{1-2x}\right)}^{-1}}{dx}}\\ & =& {\displaystyle \frac{dx{\left({(1-2x)}^{\frac{1}{2}}\right)}^{-1}}{dx}}\\ & =& {\displaystyle \frac{dx{(1-2x)}^{-\frac{1}{2}}}{dx}}\\ & =& {\displaystyle x\frac{d{(1-2x)}^{-\frac{1}{2}}}{dx}+\frac{dx}{dx}{(1-2x)}^{-\frac{1}{2}}}\\ & =& {\displaystyle x(-\frac{1}{2}){(1-2x)}^{-\frac{3}{2}}\frac{d(1-2x)}{dx}+1·\frac{1}{\sqrt{1-2x}}}\\ & =& {\displaystyle \frac{-x}{2{(1-2x)}^{\frac{3}{2}}}·(-2)+\frac{1}{{(1-2x)}^{\frac{1}{2}}}}\\ & =& {\displaystyle \frac{x+1-2x}{{(1-2x)}^{\frac{3}{2}}}}\\ & =& {\displaystyle \frac{1-x}{{(1-2x)}^{\frac{3}{2}}}\text{.}}\end{array}$$

Notes and References

These are a typed copy of Lecture 4 from a series of handwritten lecture notes for the class MATH 221 given on September 13, 2000.

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