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)

Chain Rule

100

y = (3x - 9)⁴

y' = 12(3x - 9)³

200

f(x) = sin(4x)

f'(x) = 4 cos(4x)

300

Find y' if y = (9x² + 4)^1/3

(6x)/(9x² + 4)^2/3

400

f(x) = sin³(4x)

f'(x) = 12sin²(4x) cos(4x)

500

W(x) = sin²(3x)

W'(x) = 6sin(3x) cos(3x)

Trigonometric Derivatives

100

f(x) = sec x

f'(x) = sec x tan x

200

y = 3csc(2x)

y' = -6csc(2x) cot(2x)

300

f(z) = sin x tan x

f'(z) = cos x tan x + sin x sec²x

400

y = cos²(x)

y' = -2cos(x) sin(x)

500

y = sin⁵(8x²)

y' = 5sin⁴(8x²) cos(8x²) 16x

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)