Calculus 1 Review- Finding Derivatives (pt. 2)

Inverse/Logarithms

100

Find the derivative of the function y=ln(sinx)

y'=cotx

200

Find the derivative of the function y= 3*sqr (lnx)

y'=1/3x*ln(x)^2/3

300

Find the derivative of the function y=ln(3te^-t)

y'=-t+1/t

400

Use logarithmic differentiation to find the derivative of y=(3x+2)⁴(x⁷)/sinx

dy/dx=((3x+2)⁴(x⁷)/sinx)(12/(3x+2)+7/x-cotx)

500

Find the derivative of the function y=log₂(3x-x⁴)

Hint: d/dx log_au=1/u*lna

y'=3-4x³/(3x-x⁴)ln(2)

Implicit Differentiation

100

How do you solve an implicit differentiation problem? 

- Differentiate both sides with respect to x 

- Collect dy/dx to one side and solve 

200

If x²+y²=10000, find the derivative 

dy/dx= -x/y

300

If x⁵=3x-y/x+4y, then find the derivative. 

Hint: You can rewrite from fraction form to make it easier to work through (rather than quotient rule)

dy/dx=6x⁵+20x⁴y-3/(-1-4x⁵)

400

If e^6x=sin(x+7y), then find the derivative. 

dy/dx= 6e^6x-cos(x+7y)/7(cos(x+7y))

500

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 

4xy+pi*siny=23pi         (4,3pi/2)

y=3pi/8*x+3pi

Inverse Trig

100

Find the derivative of the function y= tan^-1(x²+5x)

y'=2x+5/1+(x²+5x)²

200

What is arccsc x? 

-1/|x|sqr(x²-1)

300

Find the derivative of the function h(x)=sin^-1x/1+x? 

h'(x)=(1+x/sqr(1-x²))-sin^-1(x)/(1+x)²

400

Find the derivative of the function y=xarcsinx+sqr(1-x²)

y'=arcsinx

500

Find the derivative of the function y=ln(arccot(9x²))

y'=18x/arccot(9x²)(1+81x⁴)

Rate of Change

100

What is position? 

s(t)

200

How do you find velocity? What is speed? 

s'(t)=v(t) 

|v(t)|

300

How do you find acceleration? 

s''(t)=v'(t)=a(t)

400

A rock thrown vertically upward from the surface of the moon at a velocity of 32 m/sec reaches a height of s=32t-0.8t² meters in t sec. 

a.) Find the rock's velocity and acceleration at time t. 

b.) How long does it take the rock to reach its highest point? 

c.) How high does the rock go? 

a.) v(t)= 32-1.6t; a(t)=-1.6

b.) t=20 sec. 

c.) s(20)= 320 meters

500

At time t, the position of a body moving along the s-axis is s=-t³+12t²-45t m. 

a.) Find the body's acceleration each time the velocity is zero. 

b.) Find the body's speed each time the acceleration is zero.

c.) Find the total distance traveled by the body from t=0 to t=4.   

a.) a(3)=6 m/sec²; a(5)= -6 m/sec²

b.) |v(4)|= 3 m/sec

c.) 56m