Calculus Derivatives

Basic Derivatives

100

y = 3x²

y' = 6x

200

f(x) = 5

f'(x) = 0

300

f(x) = 6x³ + x - 2

f'(x) = 18x² + 1

400

f(x) = 3sin(x) - 2cos(x)

3cos(x) + 2sin(x)

500

s(t) = 13t⁴ + 3t²  - 6t +3

s'(t) = 52t³ + 6t - 6

Product and Quotient Rule

100

y = 3x² sin(x)

y' = 3x² cos(x) + 6x sin(x)

200

f(x) = (3x - 2x²)(5 + 4x)

-24x² + 4x + 15

300

y = (5x - 2)/(x² + 1)

y' = (-5x² + 4x + 5)/(x² + 1)²

400

h(x) = (2x + 5) / x^1/2

h'(x) = (2x - 5)/(2x^3/2)

500

Blue(x) = 3x sin(3x)

Blue'(x) = 3sin(3x) + 9x cos(3x)

Logarithmic and Exponential Derivatives

100

Find the 2^nd derivative of     y = e^x

y'' = e^x

200

f(x) = ln(x²)

2/x

300

f(x) = 3^x2

f'(x) = 2x ln3 3^x2

400

f(x) = ln (x/(x² + 1))

f'(x) = (1/x) - (2x/(x² + 1))

500

y = ln(5x²+1)⁷

70x/(5x²+1)

Applications with Derivatives

100

Find the equation of the tangent line to y = ln(x³) at x=e.

y-3=3/e (x-e) or just y = 3x/e

200

If the position of an object on the x-axis is given by x(t) = t*cos(3t) meters at time t seconds, then what is the velocity of the object when t = pi/2?

3pi/2 meters per second

300

What is the rate of change of f(t) = t²ln(2t-3) at the moment that t = 2?

8 units of f per unit of t

400

What value of x does f(x)=xe^-2x have a horizontal tangent line?

What is x = 1/2 ?

500

For what values of k does the tangent line to y = ln(x) at x=k have a y-intercept of (0,1)?

k = e²

Miscellaneous

100

The points at which a function is not differentiable.

What is a sharp corner, a cusp, and a vertical tangent line?

200

What is the definition of a derivative?

What is the limit as h approaches 0 of f(x+h)-f(x) divided by h.

300

If f is differentiable at x=c, then f is continuous at x=c. True or false?

True

400

The equation of the tangent line of f(x)=x^2 at x=2

What is y-4=4(x-2)

500

Find the equation of the normal line of f(x)=(x^2+2x)(x+1) at x=1

What is y-6=-1/11(x-1)