Derivatives

Polynomials & Exponential Functions

100

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

18x² + 1

200

Find f'(x) for f(x) = 4^x + x⁵

f'(x) = (ln4)4^x + 5x⁴

300

Differentiate y = 4π² + 3e⁴.

0

400

Find f'(x) if f(x) = x^e + x^√2- x^2π

f'(x) = ex^{e - 1} + (√2)x^{√2 - 1} - 2πx^{2π -1}

500

Find h'(t) if h(t) = ⁴√t  -  4e^t.

h'(t) = 1/(4t^3/4) - 4e^t

Product/Quotient Rule

100

What is f'(x) when f(x) = (6x³ - x)(10-20x)?

f'(x) = -480x³ + 180x² + 40x - 10

200

f = (3x - 2x²)(5 + 4x). What is df/dx?

df/dx = -24x² + 4x + 15

300

Differentiate ³√(x²) (2x - x²).

(10/3)x^2/3 - (8/3)x^5/3

400

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

(-5x² + 4x + 5) / (x² + 1)²

500

Find the derivative of (2x+5) / √x.

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

Trig Functions

100

What is the 6th derivative of sinx?

-sinx

200

What is the 445th derivative of cosx?

-sinx

300

Find g'(x) when g(x) = 3sec(x) - 10cot(x).

g'(x) = 3sec(x)tan(x) + 10csc²(x)

400

y = 5sin(x)cos(x) + 4csc(x). What is y'?

y' = 5cos²(x) - 5sin²(x) - 4csc(x)cot(x)

500

Find the derivative of sin(t) / (3 - 2cos(t)).

(3cos(t) - 2) / (3 - 2cos(t))²

Miscellaneous

100

The position of a moving car is given by the function 

s(t) = (4.5t² - 4.5t + 3) miles, where t is in seconds.

What is the car's instantaneous velocity at t = 4? (Provide units!)

31.5 mi/s

200

Find the tangent line to f(x) = 15 - 2x² at x=1.

y = -4x + 17

300

Suppose that the amount of air in a balloon after t hours is given by 

V(t) = t³ - 6t² + 35 cm³.

What is the instantaneous rate of change of volume at 5 hours? (Provide units!)

V'(5) = 15 cm³/hr

400

Determine the absolute extrema for the following function and interval.

g(t) = 2t³ + 3t² - 12t + 4  on  [-4,2]

Absolute max: 24 (occurs at critical point t = -2)

Absolute min: -28 (occurs at endpoint t = -4)

500

Suppose that the volume (AKA amount of air inside) of a balloon at any time t is given by 

V(t) = (6 * ³√t) / (4t + 1) cm³.

Determine if the air is being filled with air or being drained of air at t = 8.

V'(8) = -7/242 (negative) so volume must be decreasing --> air is being DRAINED at t = 8.

Chain Rule

100

Differentiate R(z) = √(5x - 8).

5 / (2 * √(5z-8))

200

What is f'(x) when f(x) = sin(3x² + x)?

f'(x) = (6x+1) cos(3x² + x)

300

Find f'(t) when f(t) = (2t³ + cos(t))⁵⁰

f'(t) = 50(6t² - sin(t)) (2t³ + cos(t))⁴⁹

400

Differentiate e^w4-3w2+9

(4w³ - 6w)e^w4-3w2+9

500

Differentiate cos⁴(t) + cos(t⁴).

-4sin(t)cos³(t) - 4t³sin(t⁴)