# Chapter 16: Higher Order & Trig Derivatives

### Differentiation Rules

Recall,

﻿$\frac{d}{dx}(x^{n})=nx^{n-1}$﻿

﻿$\frac{d}{dx}(k)=0$﻿

We introduce more functions

Fact: ﻿$\frac{d}{dx}(e^{x})=e^{x}$﻿

﻿$e^{x}$﻿ - Is its own derivative

#### Example

Find ﻿$\frac{dy}{dx}$﻿ if ﻿$y=xe^{x}$﻿

Product rule

﻿$\frac{dy}{dx} =\frac{d}{dx}(x)e^{x}+x\frac{d}{dx}(e^{x})$﻿

﻿$=1\cdot e^{x}+xe^{x}=(1+x)e^{x}$﻿

#### Example

Find ﻿$\frac{dr}{ds}$﻿ if ﻿$r=\frac{e^{s}}{s}$﻿

Quotient Rule

﻿$\frac{dr}{ds}=\frac{\frac{d}{ds}(e^{s})\cdot s-e^{s}\frac{d}{ds}(s)}{[s]^{2}}$﻿

﻿$=\frac{e^{s}\cdot s-e^{s}\cdot 1}{s2}= \frac{se^{s}-e^{s}}{s^{2}}$﻿

﻿$=\frac{(s-1)e^{s}}{s^{2}}$﻿

The constant ﻿$e$﻿ can be defined by

﻿$\frac{d}{dx}[\propto^{x}] =\propto^{x} \Rightarrow \propto=e$﻿

### Higher Order Derivatives

The SECOND DERIVATIVE of ﻿$y=f(x)$﻿ is ﻿$y" = \frac{d^{2}y}{dx^{2}}=f"(x)=\frac{d}{dx} f'(x)$﻿

Acceleration is the derivative of speed and the second derivative of position

In general, the nth DERIVATIVE is:

﻿$y^{(n)} =\frac{d^{n}y}{dn^{x}} = f^{(n)}(x) =\frac{d}{dx}f^{(n-1)} (x)$﻿

#### Example

Find ﻿$f'(x)$﻿ and ﻿$f''(x)$﻿ of ﻿$f(x) = x^{2}+x+1$﻿

Calculate the first derivative ﻿$f'(x)$﻿

﻿$f'(x)=2x+1$﻿

Calculate the second derivative ﻿$f''(x)$﻿

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

Example

Find ﻿$\frac{d^{3}y}{dx^{3}}$﻿ if ﻿$y=x^{3}$﻿

﻿$\frac{d^{3}y}{dx^{3}} =\frac{d}{dx}[\frac{d}{dx}[\frac{d}{dx} x^{3}]]$﻿

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

﻿$=\frac{d}{dx}\left [ 6x \right ]=6$﻿

### Trigonometric Derivatives

#### Discuss

Graphically determine the derivatives of ﻿$y= sin(x)$﻿ and ﻿$y=cos(x)$﻿

#### Example

Calculate ﻿$\frac{d^{2}y}{dx^{2}}$﻿ if ﻿$y = x^{2}sin(x)$﻿

﻿$\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}[\frac{d}{dx}[x^{2}sin(x)]]$﻿

﻿$= \frac{d}{dx}\left [ 2x\; sin(x)+x^{2}cos(x) \right ]$﻿

﻿$=2sin(x)+2xcosx+2xcos(x)-x^{2}sin(x)$﻿

﻿$=2sin(x)+4cos(x)-x^{2}sin(x)$﻿

#### Example

Find ﻿$f'(t)$﻿ if ﻿$f(t) =e^{t}cos(t)$﻿

﻿$f'(t)=\frac{d}{dt}(e^{t}) cos(t) +e^{t} \frac{d}{dx}(cos(t)$﻿

﻿$=e^{t}cost+e^{t}(-sint)$﻿

﻿$=e^{t}(cost-sint)$﻿

#### Discuss

Find the remaining derivatives

﻿$y=tan(x)$﻿ ﻿$\Rightarrow$﻿ ﻿$y= sec(x)$﻿

﻿$y=cot(x)$﻿ ﻿$\Rightarrow$﻿ ﻿$y=csc(x)$﻿

﻿$y=tax(x)=\frac{sin(x)}{cos(x)}$﻿ ﻿$\Rightarrow$﻿ ﻿$\frac{dy}{dx}= \frac{cos(x)cos(x)+sin(x)sin(x)}{[cos(x)]^{2}}$﻿ ﻿$= sec^{2}(x)$﻿

﻿$y=cot(x)=\frac{cos(x)}{sin(x)}$﻿ ﻿$\Rightarrow$﻿ ﻿$\frac{dy}{dx}= \frac{-sin(x)sin(x)-cos(x)cos(x)}{[sin(x)]2}$﻿ ﻿$= -csc^{2}(x)$﻿

﻿$y=sec(x)=\frac{1}{cos(x)}$﻿ ﻿$\Rightarrow$﻿ ﻿$\frac{dy}{dx}=\frac{0-1(-sin(x))}{cos^{2}(x)}$﻿ ﻿$=tan(x)sec(x)$﻿

﻿$y=csc(x)=\frac{1}{sin(x)}$﻿ ﻿$\Rightarrow$﻿ ﻿$\frac{dy}{dx}=\frac{0-1.cos(x)}{sin^{2}(x)}$﻿ ﻿$= -cot(x)\cdot csc(x)$﻿

We can derive the remaining trig derivatives from: ﻿$sin(x)$﻿ and ﻿$cos(x)$﻿

The only commonly used formula is:

﻿$\frac{d}{dx} sec(x)=tan(x)sec(x)$﻿

## Summary

﻿$\frac{d}{dx}[ex]=e^{x}$﻿

﻿$\frac{d}{dx}[sin(x)]=cos(x)$﻿

﻿$\frac{d}{dx}[cos(x)]=-sin(x)$﻿

﻿$\frac{d^{n}y}{dx^{n}}=\frac{d}{dx}[\frac{d^{n-1}y}{dx^{n-1}}]$﻿

﻿$f^{(n)}(x)=\frac{d}{dx}[f^{(n-1)}(x)]$﻿