Chapter 3: Chain Rule, Implicit Differentiation... Review

Chain Rule

100

Differentiate

y=sqrt(2x)

1/(sqrt(2x)

200

Differentiate

y=sin^2x

y'=2sinxcosx

300

Differentiate

y=(x^2+4)^(3/2)

y'=3xsqrt(x^2+4)

400

Differentiate

y=ln(tan^2x)

2sec^2x/(tanx)

500

Differentiate

f(x)=e^(x^2cosx)

f'(x)=-xe^(x^2cosx)(xsinx-2cosx)

Implicit Differentiation

100

Differentiate

x=y^5+y

dy/dx=1/(5y^4+1)

200

Differentiate

xy=y

dy/dx=y/(1-x)

300

Find y'.

xe^y=4x^2

y'=(8x-e^y)/(xe^y)

400

Differentiate

tan(xy)=x

dy/dx=(1-ysec^2(xy))/(xsec^2(xy))

500

Find y''.

y^3+y=x^2

y''=(2(3y^2+1)^2-24x^2y)/(3y^2+1)^3

Inverse Trig

100

Differentiate

y=arcsin(2x)

y'=2/(sqrt(1-4x^2)

200

Differentiate

f(x)=arccos(2x-1)

f'(x)=-2/(sqrt(1-(2x-1)^2

300

Differentiate.

g(r)=4tan^-1(r/2)+r

g'(r)=8/(r^2+4)+1

400

Differentiate.

y=sqrt(1-x^2)(sin^-1x)

y'=1-(xsin^-1x)/(sqrt(1-x^2)

500

Differentiate.

cot^-1(xy)=y

dy/dx=y/(-1-x-x^2y^2)

Exponentials and Logarithms

100

Differentiate

y=x5^x

y'=5^x+ln(5)x5^x

200

Differentiate

f(x)=6^(x^2)

f'(x)=2ln(6)x6^(x^2)

300

Differentiate

g(t)=t^2-log2t

g'(t)=2t-1/(ln10t)

400

Differentiate

f(x)=log_2(sin(2x))

f'(x)=2cot(2x)/(ln2)

500

Differentiate.

log_3(x+y)=8x

dy/dx=8ln3(x+y)-1

Tangent Lines and Applications

100

Find the slope of the tangent line of f(x) when x=0.

f(x)=(2x-1)^3

f'(x)=6(2x-1)^2

f'(0)=6

200

Find the slope of the tangent line of f(x) when x=e.

f(x)=x^x

f'(x)=x^x(lnx+1)

f'(e)=2e^e

300

Consider the curve below. Find all the points where the tangent line to the curve has a slope of 1/2.

y^2 = 2+xy

(0,sqrt2)

(0,-sqrt2)

400

Consider the function f=y y>0 whose curve is given by the equation below. Write an equation for the line tangent to the curve at the point:

(0,sqrt3)

2y^2-6=ysinx

y=sqrt3+1/4x

500

Consider the curve below. Write an equation for each horizontal tangent line the curve has. (You can use a graphing calculator to solve for y)

2y^3+6x^2y-12x^2+6y=1

y=0.165