Calculus Derivative Guide

Derivative of sin x

^{cos x}

Derivative of cos x

-sin x

Derivative of tan x

sec² x

Derivative of cot x

-csc² x

Derivative of sec x

sec x tan x

Derivative of csc x

-cot x csc x

Derivative of e^x

e^x

Product Rule

f(x)*g’(x) + g(x)*f’(x)

Quotient Rule f(x) / g(x)

(g(x)*f’(x) - f(x)*g’(x)) / (g(x))²

Power Rule (derivative of xⁿ)

nx^n-1

• x³=3x²

Constant Rule (derivative of a constant)

0

• 5=0

• pi=0

Constant Multiple Rule (cf(x))

cf’(x)

• 5x⁴=20x³

Sum and Difference Rules (derivative of f(x) ± g(x))

f’(x) ± g’(x)

• x⁴+3x²-1 = 4x³+6x

Derivative of sin^-1 x

1 / sqrt(1 - x²)

Derivative of cos^-1 x

-1 / sqrt(1 - x²)

Derivative of tan^-1 x

1 / 1 + x²

Derivative of cot^-1 x

-1 / 1 + x²

Derivative of sec^-1 x

1 / |x|*sqrt(x² - 1)

Derivative of csc^-1 x

-1 / |x|*sqrt(x² - 1)

Chain Rule f(g(x))

f’(g(x))*g’(x)

Derivative of ln x or log_e x

1 / x

Derivative of log_a x

1 / x*ln n

Implicit Rule (dy/dx (x²+y²=25)

2x + 2y*y’ = 0

y’ = -x / y

Expanding Rule (3ײ + 1)²

(3ײ + 1)*(3ײ + 1)

(9×⁴ + 6ײ + 1)

y’ = 36׳ + 12x

Outside-Inside Rule (3ײ + 1)²

(3ײ + 1)²

y’ = 2(3ײ + 1)*6x

y’ = 36׳ + 12x

U - Substitution (3ײ + 1)²

(3ײ + 1)²

y’ = 2u * 6x

y’ = 2(3ײ + 1)² * 6x

y’ = 36׳ + 12x

|x| is equivalent to

Sqrt(x²)

Derivative of a^x

a^x * lna

Derivaticve of lne^x or e^lnx

x

y = log_b x is equivilant to

b^y = x

y = log₁₀ x is equivilant to

y = log x

y = log_e x is equivilant to

y = ln x

When log_b(xy)

log_b x + log_b y or ln x + ln y becauce b^m * bⁿ = b^m+n

When log_b(x/y)

log_b x - log_b y or ln x - ln y becauce b^m/bⁿ = b^m-n

When log_b x^p

P * log_b x or P * ln x becauce (b^m)ⁿ = b^m*n

Derivative lny = any function

y*(derivative of that function