Chapter 15: Differentiation Rules

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

ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)


Fact: If f(x)=xnf(x)=x^{n} then f(x)=nxn1f'(x)=nx^{n-1}

ddxxn=nxn1\frac{d}{dx}x^{n}=nx^{n-1}


Example \rightarrow Find f(x)f'(x) if f(x)=3x2+2.x+1f(x) =3x^{2}+2.x+1

ddxf(x)=ddx[3x2+2x+1]\frac{d}{dx}f(x)=\frac{d}{dx}[3x^{2}+2x+1]


Addition =ddx[3x2]+ddx[2x]+ddx[1]=\frac{d}{dx}[3x^{2}] +\frac{d}{dx}[2x] +\frac{d}{dx}[1]


Scaling =3ddx[x2]+2ddx[x]+ddx[1]=3\frac{d}{dx}[x^{2}]+ 2\frac{d}{dx}[x]+\frac{d}{dx}[1]


Power =6x+2.1+0=6x+2= 6x+2.1+0 =6x+2


Example \rightarrow Find dydx\frac{dy}{dx} if y=42xx3y=4-2x-x^{-3}

dydx=02(3)x31\frac{dy}{dx}=0-2-(-3)x^{-3-1}

=2+3x4=2+3x^{-4}


Example \rightarrow Find drdθ\frac{dr}{d\theta} if r=12θ4θ3+1θ4r =\frac{12}{\theta} -\frac{4}{\theta^{3}}+\frac{1}{\theta^{4}}

Re-write with negative exponents

r=12θ14θ3+θ4r=12 \theta^{-1}-4 \theta^{-3}+\theta^{-4}

drdθ=12θ2+12θ44θ5\frac{dr}{d\theta}=-12 \theta^{-2}+12 \theta^{-4}-4\theta^{-5}


Example \rightarrow Find all the points (x,y)(x,y) where y=x33x2y=x^{3}-3x-2 has a horizontal tangent line.

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

dydx=3x23=03(x21)=0\frac{dy}{dx}= 3x^{2}-3 =0 \Leftrightarrow 3(x^{2}-1)=0

x=±1\Leftrightarrow x=\pm1

Thus, there are horizontal tangent lines at:

x=1x=-1 and x=1x=1

Evaluate for yy

y=(1)33(1)2=1+32=0y=(-1)^{3}-3(-1)-2=-1+3-2=0

y=(1)33(1)2=4y=(1)^{3}-3(1)-2=-4

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

Example \rightarrow Find AA such that

g(x)=Ax  x<2g(x)=Ax\; x<2

g(x)=x23x  x2g(x)=x^{2}-3x\; x \geq 2 is differentiable everywhere

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

g+(2)=2.23=1g'+(2)=2.2-3=1 

g(2)=Ag'(2)=A

g+(x)=2x3g'+(x)=2\cdot x-3 at x=2x=2

Thus, A=1A=1  and

g(x)=x  x<2g(x) = {x\; x< 2}

g(x)=x23x  x2g(x) = x^{2}-3x\; x\geq 2

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

dhdx=dfdgg+fdgdx\frac{dh}{dx}=\frac{df}{dg}g+f \frac{dg}{dx}

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

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

Example \rightarrow Find f(t)f'(t) if f(t)=t21t2+t2f(t)= \frac{t^{2}-1}{t^{2}+t-2}

f(t)=ddt(t21)[t2+t2][t21]ddt(t2+t2)(t2+t2)2f'(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}}

=2t[t2+t2][t21](2t+1)(t2+t2)2=\frac{2t[t^{2}+t-2]-[t^{2}-1](2t+1)}{(t^{2}+t-2)^{2}}

=2t3+2t24t2t3+2t+1(t2+t2)2=\frac{{ \cancel{2t^3} + 2t^2-4t- \cancel{2t^3}+2t+1}}{(t^2+t-2)^2}

=t22t+1(t2+t2)2=\frac{t^{2}-2t+1}{(t^{2}+t-2)^{2}}


Example \rightarrow Find dydx\frac{dy}{dx} if y=(x2+1)(3x)y=(x^{2}+1)(3x)

dydx=[ddx(x2+1)](3x)+(x2+1)[ddx(3x)]\frac{dy}{dx}=[\frac{d}{dx}(x^{2}+1)](3x)+(x^{2}+1)[\frac{d}{dx}(3x)]

=2x(3x)+(x2+1)(3)=9x2+3=2x(3x)+(x^{2}+1)\cdot (3)=9x^{2}+3


Summary

Constants

f(x)=kf(x)=0f(x)=k \Rightarrow f'(x)=0

Scaling

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

Addition

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

Powers

f(x)=xnf(x)=nxn1f(x)=xn \Rightarrow f'(x)=nx^{n-1}

Multiplication

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

Quotients

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









































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