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 n^th 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

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

$\frac{d}{dx}[sin(x)]=cos(x)$

$\frac{d}{dx}[cos(x)]=-sin(x)$

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

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