Finding Derivatives

Basic Derivative Information and Power Rule

100

The derivative calculates the ________ of a function.

Slope

200

d/dx (5)

0

300

d/dx x^2

2x

400

d/dx (3x^2-x+3)

6x-1

500

d/dx(1/sqrt(x))

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

Chain Rule

100

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

g′(x)f′(g(x))

200

d/dx (3x+1)^2

6(3x+1)

300

d/dx sin(4x^2)

8xcos(4x^2)

400

Differentiate 

y=sqrt(13x^2-5x+8)

y′ =(26x-5)/ (2sqrt(13x²-5x+8))

500

Differentiate 

y=3tan(sqrt(x))

y′ =(3sec^2(sqrt(x)))/ (2sqrt(x))

Product Rule

100

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

f′(x)g(x)+ g′(x)f(x)

200

What is f'(x) if 

f(x)=x^2sinx

2xsinx+ x^2cosx

300

Differentiate 

y=x^3lnx

y′ =x^2(1+3lnx)

400

Differentiate 

y=e^(-x^2)cos2x

y′ =−2xe^(−x²) cos2x−2e^(−x²)sin2x

500

Differentiate 

y=x^2sin^3(5x)

y′ =xsin^2(5x)[15xcos(5x)+2sin(5x)]

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

Differentiate 

y= (1+lnx) / (x^2-lnx)

y′= [(1/x)-x-2xlnx] / (x^2-lnx)^2

400

What is f′(x) if 

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

f′(x)= [4x(x²-1)²(x²+2)] / (x²+1)²

500

Differentiate 

y= (x^3lnx)/(x+2)

y′ = [x^2(2xlnx+6lnx+x+2)]/ (x+2)^2

Trig Derivatives

100

d/dx cosx

-sinx

200

Differentiate 

y=tan(x)

y′ =sec²(x)

300

Differentiate 

y=csc(x)

y′ =-csc(x)cot(x)

400

d/dx sin(2x)

2cos(2x)

500

d/dx sec(5x)=

5sec(5x)tan(5x)