Chapter 16: Higher Order & Trig Derivatives

Differentiation Rules

Recall,

ddx(xn)=nxn1\frac{d}{dx}(x^{n})=nx^{n-1}

ddx(k)=0\frac{d}{dx}(k)=0

We introduce more functions


Fact: ddx(ex)=ex\frac{d}{dx}(e^{x})=e^{x}

exe^{x} - Is its own derivative


Example

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

Product rule

dydx=ddx(x)ex+xddx(ex)\frac{dy}{dx} =\frac{d}{dx}(x)e^{x}+x\frac{d}{dx}(e^{x}) 

=1ex+xex=(1+x)ex=1\cdot e^{x}+xe^{x}=(1+x)e^{x} 


Example

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

Quotient Rule

drds=dds(es)sesdds(s)[s]2\frac{dr}{ds}=\frac{\frac{d}{ds}(e^{s})\cdot s-e^{s}\frac{d}{ds}(s)}{[s]^{2}}

=esses1s2=sesess2=\frac{e^{s}\cdot s-e^{s}\cdot 1}{s2}= \frac{se^{s}-e^{s}}{s^{2}}

=(s1)ess2=\frac{(s-1)e^{s}}{s^{2}}


The constant ee can be defined by

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


Higher Order Derivatives

The SECOND DERIVATIVE of y=f(x)y=f(x) is y"=d2ydx2=f"(x)=ddxf(x)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)=dnydnx=f(n)(x)=ddxf(n1)(x)y^{(n)} =\frac{d^{n}y}{dn^{x}} = f^{(n)}(x) =\frac{d}{dx}f^{(n-1)} (x)


Example

Find f(x) f'(x) and f(x)f''(x) of f(x)=x2+x+1f(x) = x^{2}+x+1

Calculate the first derivative f(x)f'(x)

f(x)=2x+1f'(x)=2x+1

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

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


Example

Find d3ydx3\frac{d^{3}y}{dx^{3}} if y=x3y=x^{3}

d3ydx3=ddx[ddx[ddxx3]]\frac{d^{3}y}{dx^{3}} =\frac{d}{dx}[\frac{d}{dx}[\frac{d}{dx} x^{3}]]

=ddx[ddx[3x2]]=\frac{d}{dx}[\frac{d}{dx}[ 3x^{2}]]

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


Trigonometric Derivatives

Discuss

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


Example

Calculate d2ydx2\frac{d^{2}y}{dx^{2}} if y=x2sin(x)y = x^{2}sin(x)

d2ydx2=ddx[ddx[x2sin(x)]]\frac{d^{2}y}{dx^{2}} = \frac{d}{dx}[\frac{d}{dx}[x^{2}sin(x)]]

=ddx[2xsin(x)+x2cos(x)]= \frac{d}{dx}\left [ 2x\; sin(x)+x^{2}cos(x) \right ]

=2sin(x)+2xcosx+2xcos(x)x2sin(x)=2sin(x)+2xcosx+2xcos(x)-x^{2}sin(x)

=2sin(x)+4cos(x)x2sin(x)=2sin(x)+4cos(x)-x^{2}sin(x)


Example

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

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

=etcost+et(sint)=e^{t}cost+e^{t}(-sint)

=et(costsint)=e^{t}(cost-sint)


Discuss

Find the remaining derivatives

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

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

y=tax(x)=sin(x)cos(x)y=tax(x)=\frac{sin(x)}{cos(x)} \Rightarrow dydx=cos(x)cos(x)+sin(x)sin(x)[cos(x)]2\frac{dy}{dx}= \frac{cos(x)cos(x)+sin(x)sin(x)}{[cos(x)]^{2}} =sec2(x)= sec^{2}(x)

y=cot(x)=cos(x)sin(x)y=cot(x)=\frac{cos(x)}{sin(x)} \Rightarrow dydx=sin(x)sin(x)cos(x)cos(x)[sin(x)]2\frac{dy}{dx}= \frac{-sin(x)sin(x)-cos(x)cos(x)}{[sin(x)]2} =csc2(x)= -csc^{2}(x)

y=sec(x)=1cos(x)y=sec(x)=\frac{1}{cos(x)} \Rightarrow dydx=01(sin(x))cos2(x)\frac{dy}{dx}=\frac{0-1(-sin(x))}{cos^{2}(x)} =tan(x)sec(x)=tan(x)sec(x)

y=csc(x)=1sin(x)y=csc(x)=\frac{1}{sin(x)} \Rightarrow dydx=01.cos(x)sin2(x)\frac{dy}{dx}=\frac{0-1.cos(x)}{sin^{2}(x)} =cot(x)csc(x)= -cot(x)\cdot csc(x)


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

The only commonly used formula is:

ddxsec(x)=tan(x)sec(x)\frac{d}{dx} sec(x)=tan(x)sec(x)

Summary

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

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

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

dnydxn=ddx[dn1ydxn1]\frac{d^{n}y}{dx^{n}}=\frac{d}{dx}[\frac{d^{n-1}y}{dx^{n-1}}]

f(n)(x)=ddx[f(n1)(x)]f^{(n)}(x)=\frac{d}{dx}[f^{(n-1)}(x)]


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