Calculus Derivatives - BBS

Polynomials

100

What is the derivative of y = 3x² ?

y' = 6x

200

Find the derivative of f(x) = 5

0

300

Find the derivative of f(x) = 6x³ + x - 2

18x² + 1

400

Find the derivative of y=-2/(x⁷).

y'=14x^-8

500

Find the derivative of g(x)=2x^1/2.

g'(x)=x^-1/2

Products

100

Find the derivative of f(x)=(2x-2)(4x+5).

f'(x)=2(4x+5)+4(2x-2)

200

What is the derivative of g(x)=xsin(x)?

g'(x)=sin(x)+xcos(x)

300

What is the derivative of h(x)=x²e^x?

h'(x)=2xe^x+x²e^x

400

Find the derivative of f(x)=sec(x)cos(x).

f'(x)=0

500

What is the derivative of y=e^2x?

y'=2e^2x

Quotients

100

What is the derivative of h(x)=x²/(x+1)?

h'(x)=((2x)(x+1)-x²)/(x+1)²

200

What is the derivative of y=6x²/7?

y'=12/7 x

300

Find the derivative of f(x)=sin(x)/e^x.

f'(x)=(e^xcos(x)-e^xsin(x))/e^2x

400

Find the derivative of f(x)=(3x+x²)/x

f'(x)=1

500

Find the derivative of h(x)=(1-sin(x))/(1+cos(x)).

h'(x)=((1+cos(x))(-cos(x))-(1-sin(x))(-sin(x)))/(1+cos(x))²

Trigonometric Functions

100

Find the derivative of y=sin(x).

y'=cos(x)

200

What is the derivative of f(t)=cos(t)?

f'(t)=-sin(t)

300

Find the derivative of y=-tan(x).

y'=-sec²(x)

400

What is the derivative of y=csc(x)?

y'=-csc(x)cot(x)

500

What is the derivative of y=-1/6 cot(x)?

y'=1/6 csc²(x)

Formulas

100

What is the derivative of a function y=xⁿ?

y'=nx^n-1

200

The formula for the product rule on h(x)=f(x)g(x).

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

300

Formula for average rate of change on an interval [a,b]

(f(b)-f(a))/(b-a)

400

The limit definition of the derivative

f'(x)=lim h ->0 (f(x+h)-f(x))/h

500

The formula for the quotient rule on h(x)=f(x)/g(x)

h'(x)=(g(x)f'(x)-f(x)g'(x))/(g(x))²