Topic 3.1 and 3.2 - Chain Rule

Polynomials

100

Derive and fully simplify (don't expand)

f(x)=(3x^2+2)^5

30x(3x^2+2)^4

200

Derive and fully simplify (No negative exponents)

sqrt(x^3+1)

(3x^2)/(2sqrt(x^3+1)

300

Derive and fully simplify (No negative exponents)

f(x)=1/((x^2+4)^(3/2)

(-3x)/((x^2+4)^(5/2)

400

Determine the slope of the tangent of f(x) at x=1

f(x)=(3x^2+2)^2

60

500

Find the equation of the tangent line of f(x) at the point where x=1. Keep it in point-slope form

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

y-36=108(x-1)

Trigonometry

100

Derive

f(x)=sin(3x)

3cos(3x)

200

Derive

f(x)=cos(2x^2+lnx)

f'(x)=-sin(2x^2+lnx)(4x+1/x)

300

Derive

f(x)=tan(x^3)

f'(x)=sec^2(x^3)3x^2

400

Derive

h(x)=sin^2(2x+1)

4sin(2x+1)cos(2x+1)

500

Derive

f(x)=(x^2+1)(csc(3x))

f'(x)=2xcsc(3x)-3(x^2+1)csc(3x)cot(3x)

Logarithms

100

Derive

f(x)=ln(3x^2+5x)

(6x+5)/(3x^2+5x)

200

Derive 

f(x)=ln(e^(2x)+4)

f'(x)=(2e^(2x))/(e^(2x)+4)

300

Derive (simplify the trigonometry)

h(x)=ln(sin(x^2))

h'(x)=2xcot(x^2)

400

Derive (no negative exponents)

p(x)=ln(x^3+sqrt(x))

p'(x)=(3x^2+1/(2sqrt(x)))/(x^3+sqrt(x)

500

Derive (use logarithm properties, no quotient rule)

q(x)=ln((x^2+1)/(x^3+2))

(2x)/(x^2+1)-(3x^2)/(x^3+2)

Exponentials

100

Derive

f(x)=e^(2x)

2e^(2x)

200

Derive

e^(3x^2+4x)

e^(3x^2+4x)(6x+4)

300

Derive

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

2sec^2(x)e^(tan(2x))

400

Surprise! It's an anime clue

Name of Gon's dad from Hunter x Hunter

Ging Freecss

500

Derive (simplify and factor)

h(x)=x^3e^(x^2)

h'(x)=e^(x^2)(3x^2+2x^4)

Tabular

100

Given

h(x)=f(g(x))

h'(4)

28

200

Given 

h(x)=g(f(x))

h'(4)

56

300

Given 

h(x)=f(g(x+2))

h'(1)

12

400

Given 

h(x)=f(g(2x))

h'(2)

56

500

Given 

h(x)=g(f(x/5))

h'(5)

2/5