Differentiation Rules

Basic Rules

100

What is the product rule?

d/dx (f(x) * g(x)) = f'(x)g(x) + f(x)g'(x)

200

What is the quotient rule for derivatives?

d/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / (g(x))^2

OR

[ho*d(hi) - hi*d(ho)] / (ho*ho

300

What is the chain rule for derivatives?

d/dx (f(g(x)) = f'(g(x))*g'(x)

400

f(x)=(4x^2-3x+2)^-2

f'(x)= 

f'(x)=(-2(8x-3))/(4x^2-3x+2)^3

500

f(x)=(3sinx+lnx)/(x^3+2x)

f'(x)=

f'(x)=((x^3+2x)(3cosx+1/x)-(3sinx+lnx)(3x^2+2))/((x^3+2x)^2

Trig & Inverse Trig Functions

100

d/dx(cosx)=

-sinx

200

d/dx(-sin^2x)=

-2sinxcosx

300

d/dx(cos^3(4x))

-12cos^2(4x)sin(4x)

400

If f(x)=x/tanx, find f'(π/4)

1-pi/2

500

d/dx(-secxcotx)=

cscxcotx

Exponential & Logarithmic Functions

100

d/dx(ln(x))

1/x

200

d/dx(b^x)

b^x*ln(b)

300

d/dx(2xe^(2x))=

4xe^(2x)+2e^(2x)

400

If f(x)=3^cosx, find f'(π/2)

f'(pi/2)=-ln3

500

Only need it in one term, not completely simplfied

d/dx(e^(3ln(4x^2+1)))

(24xe^3ln(4x^2+1))/(4x^2+1)

Potpourri

100

What rule would you use to find the derivative of cos^4(x)?

A) chain rule, B) product rule, C) trig product rule, D) L'hopital's rule

A) chain rule

200

Given x^3-y^2=0, evaluate dy/dx at the point (1,-1)

dy/dx|(1,-1) = -3/2

300

If f(x)=(x^2-2x-1)^(2/3), then f'(0)=

4/3

400

-7/(sqrt(1-49x^2))+(6lnx)/x

-7/(sqrt(1-49x^2))+(6lnx)/x

500

d/dx((-12x^4-16x^3+12x^2)/(4x^6))

(96x^9+192x^8-192x^7)/(4x^12)