Differential Calculus

Question 1

Power Rule

Answer 1

d/dx xⁿ = nx^{n - 1}

ex. d/dx 2x³ = 2(3)x^{(3 - 1)}

Question 2

Product Rule

Answer 2

d/dx (fg) = fg’ + gf’

ex. d/dy (3x²2x³) = 3x²(d/dx (2x³)) + 2x³(d/dx (3x²))

Question 3

Quotient Rule

Answer 3

d/dx (u/v) = vu’ - uv’ / v²

ex. d/dx (3x²/ln x) = ((ln x)(6x) - (3x²)(1/x)) / ln²x

Question 4

Chain Rule

Answer 4

d/dx f(g(x)) = f’(g(x))g’(x)

ex. d/dx(sin(ln x)) = cos(ln x)(1/x)

Question 5

Derivative of Sine

Answer 5

d/dx sin x = cos x

Question 6

Derivative of Cosine

Answer 6

d/dx cos x = -sin x

Question 7

Derivative of Tangent

Answer 7

d/dx tan x = sec² x

Question 8

Derivative of Cotangent

Answer 8

d/dx cot x = -csc² x

Question 9

Derivative of Secant

Answer 9

d/dx sec x = (sec x)(tan x)

Question 10

Derivative of Cosecant

Answer 10

d/dx csc x = (-csc x)(cot x)

Question 11

Derivative of e^x

Answer 11

d/dx e^x = e^x

Question 12

Derivative of a^x

Answer 12

d/dx a^x = a^x ln a

Question 13

Derivative of ln x

Answer 13

d/dx ln x = 1/x

Question 14

Derivative of Arcsine

Answer 14

d/dx sin^-1 x = 1 / sqrt(1 - x²)

Question 15

Derivative of Arctangent

Answer 15

d/dx tan^-1 x = 1 / (1 + x²)

Question 16

Derivative of Arcsecant

Answer 16

d/dx sin^-1 x = 1 / |x|sqrt(x² - 1)

Question 17

Derivative of a Constant Term

Answer 17

d/dx a = 0