Calculus Derivatives

Basic Derivatives

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 f(x) = 3sinx - 2cosx

3cosx + 2sinx

Product Rule

100

Define the Product Rule using f(x)g(x)

f′(x)g(x)+ g′(x)f(x)

200

Find the derivative of y = 3x² sin x

3x² cos x + 6x sin x

300

f(x) = (3x - 2x²)(5 + 4x)

-24x² + 4x + 15

400

f(x)=x²sinx, what is f′(x)?

2xsinx+ x²cosx

Chain Rule

100

Define the Chain Rule for f(g(x))

g′(x)f′(g(x))

200

Find y' if y = (3x - 9)⁴

12(3x - 9)³

300

Find the derivative of f(x) = sin 4x

4 cos 4x

400

Find y' if y = (9x² + 4)^1/3

(6x)/(9x² + 4)^2/3

Quotient Rule

100

Define the Quotient Rule using f(x)/g(x)

[g(x)f′(x) - f(x)g′(x)]/ (g(x))²

200

Differentiate y= 2/(x+1)

y′ = -2/ (x+1)²

300

Find the derivative of y = (5x - 2)/(x² + 1)

(-5x² + 4x + 5)/(x² + 1)²

400

Find the derivative of f(x) = (2x + 5)/(x)^1/2

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

Combining Rules

100

d/dx sin(2x)cos(2x)

-2sin²(2x)+2cos²(2x)

200

Differentiate f(x) = [(3x-1)/(x²+3)]

*Leave numerator in factored form

2(3x-1)(-3x²+2x+9)] / (x²+3)³

300

f(x) = sin³ 4x Find f'(x)

12sin² 4x cos 4x

400

Differentiate y=x²sin³(5x)

y′ =xsin²(5x)[15xcos(5x)+2sin(5x)]