# Chapter 15: Differentiation Rules

Fact: If ﻿$h(x)= f(x)+g(x)$﻿ then ﻿$h'(x)=f'(x)+g'(x)$﻿

﻿$\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$﻿

Fact: If ﻿$f(x)=x^{n}$﻿ then ﻿$f'(x)=nx^{n-1}$﻿

﻿$\frac{d}{dx}x^{n}=nx^{n-1}$﻿

Example ﻿$\rightarrow$﻿ Find ﻿$f'(x)$﻿ if ﻿$f(x) =3x^{2}+2.x+1$﻿

﻿$\frac{d}{dx}f(x)=\frac{d}{dx}[3x^{2}+2x+1]$﻿

Addition ﻿$=\frac{d}{dx}[3x^{2}] +\frac{d}{dx}[2x] +\frac{d}{dx}$﻿

Scaling ﻿$=3\frac{d}{dx}[x^{2}]+ 2\frac{d}{dx}[x]+\frac{d}{dx}$﻿

Power ﻿$= 6x+2.1+0 =6x+2$﻿

Example ﻿$\rightarrow$﻿ Find ﻿$\frac{dy}{dx}$﻿ if ﻿$y=4-2x-x^{-3}$﻿

﻿$\frac{dy}{dx}=0-2-(-3)x^{-3-1}$﻿

﻿$=2+3x^{-4}$﻿

Example ﻿$\rightarrow$﻿ Find ﻿$\frac{dr}{d\theta}$﻿ if ﻿$r =\frac{12}{\theta} -\frac{4}{\theta^{3}}+\frac{1}{\theta^{4}}$﻿

Re-write with negative exponents

﻿$r=12 \theta^{-1}-4 \theta^{-3}+\theta^{-4}$﻿

﻿$\frac{dr}{d\theta}=-12 \theta^{-2}+12 \theta^{-4}-4\theta^{-5}$﻿

Example ﻿$\rightarrow$﻿ Find all the points ﻿$(x,y)$﻿ where ﻿$y=x^{3}-3x-2$﻿ has a horizontal tangent line.

Horizontal tangent ﻿$\Leftrightarrow \frac{dy}{dx}=0$﻿

﻿$\frac{dy}{dx}= 3x^{2}-3 =0 \Leftrightarrow 3(x^{2}-1)=0$﻿

﻿$\Leftrightarrow x=\pm1$﻿

Thus, there are horizontal tangent lines at:

﻿$x=-1$﻿ and ﻿$x=1$﻿

Evaluate for ﻿$y$﻿

﻿$y=(-1)^{3}-3(-1)-2=-1+3-2=0$﻿

﻿$y=(1)^{3}-3(1)-2=-4$﻿

Thus, the tangents at ﻿$(-1,0)$﻿ and ﻿$(1,-4)$﻿ are horizontal

Example ﻿$\rightarrow$﻿ Find ﻿$A$﻿ such that

﻿$g(x)=Ax\; x<2$﻿

﻿$g(x)=x^{2}-3x\; x \geq 2$﻿ is differentiable everywhere

Need ﻿$g'+(2)=g'-(2)$﻿ to be differentiable

﻿$g'+(2)=2.2-3=1$﻿

﻿$g'(2)=A$﻿

﻿$g'+(x)=2\cdot x-3$﻿ at ﻿$x=2$﻿

Thus, ﻿$A=1$﻿ and

﻿$﻿g(x) = {x\; x< 2﻿}$﻿

﻿$﻿g(x) = x^{2}-3x\; x\geq 2$﻿

Fact: If ﻿$h(x)=f(x)g(x)$﻿ then ﻿$h'(x)=f'(x)g(x)+f(x)g'(x)$﻿

﻿$\frac{dh}{dx}=\frac{df}{dg}g+f \frac{dg}{dx}$﻿

Fact: If ﻿$h(x)=\frac{f(x)}{g(x)}$﻿ then ﻿$h'(x)= \frac{f'(x)g(x)-f(x)g'(x}{[g(x)]^{2}}$﻿

﻿$\frac{dh}{dx}= \frac{\frac{df}{dx}g-f\frac{dg}{dx}}{\left [ g \right ]^{2}}$﻿

Example ﻿$\rightarrow$﻿ Find ﻿$f'(t)$﻿ if ﻿$f(t)= \frac{t^{2}-1}{t^{2}+t-2}$﻿

﻿$f'(t)=\frac{\frac{d}{dt}(t^{2}-1)[t^{2}+t-2]-[t^{2}-1]\frac{d}{dt}(t^{2}+t-2)}{(t^{2}+t-2)^{2}}$﻿

﻿$=\frac{2t[t^{2}+t-2]-[t^{2}-1](2t+1)}{(t^{2}+t-2)^{2}}$﻿

﻿$=\frac{{ \cancel{2t^3} + 2t^2-4t- \cancel{2t^3}+2t+1}}{(t^2+t-2)^2}$﻿

﻿$=\frac{t^{2}-2t+1}{(t^{2}+t-2)^{2}}$﻿

Example ﻿$\rightarrow$﻿ Find ﻿$\frac{dy}{dx}$﻿ if ﻿$y=(x^{2}+1)(3x)$﻿

﻿$\frac{dy}{dx}=[\frac{d}{dx}(x^{2}+1)](3x)+(x^{2}+1)[\frac{d}{dx}(3x)]$﻿

﻿$=2x(3x)+(x^{2}+1)\cdot (3)=9x^{2}+3$﻿

## Summary

### Constants

﻿$f(x)=k \Rightarrow f'(x)=0$﻿

### Scaling

﻿$f(x)=c \cdot g(x) \Rightarrow f'(x)= c\cdot g'(x)$﻿

﻿$h(x)=f(x)+g(x) \Rightarrow h'(x)=f'(x)+g'(x)$﻿

### Powers

﻿$f(x)=xn \Rightarrow f'(x)=nx^{n-1}$﻿

### Multiplication

﻿$h(x)=f(x)g(x) \Rightarrow h'(x)=f'(x)g(x)+f(x)g'(x)$﻿

### Quotients

﻿$h(x)=\frac{f(x)}{g(x)} \Rightarrow h'(x) = \frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]2}$﻿