Constant Rule Power Rule

The Sum and Difference RuleProduct RuleQuotient Rule

Chain RuleTrig Functions

Exponential Function Rules( e^x ) Logarithmic Rule

The derivative of a constant function is 0.

d/dx(c) = 0

d/dx[x^n] = nx^n-1

Example:

f(x) = x^3 g(x)= 5x^2 h(x) = 2x^5

f’(x) = 3x^2 g’(x)= 10x h’(x) = 10x^4

Sum Rule: d/dx[ f(x)+g(x) ] = f’(x) + g’(x)

Difference Rule d/dx[ f(x) - g(x) ] = f’(x) - g’(x)

Examples:f(x) = x^3 - 4x + 5f’(x) = 3x^2 – 4

g(x) = -(x^4/x) + 3x^3 – 2x g’(x) = -2x^3 + 9x^2 - 2

d/dx[ f(x)g(x) ] = f(x)g’(x) + g(x)f’(x)

Example:h(x) = (3x - 2x^2)(5 + 4x)

(3x - 2x^2) d/dx[5 + 4x] + (5 + 4x) d/dx[3x - 2x^2]

(12x – 8x^2) + (15 – 8x – 16x^2)

h’(x) = -24x^2 + 4x + 15

d/dx[ f(x) / g(x) ] = g(x)f’(x) - f(x)g’(x) / g(x)^2

Example: y = (5x – 2) / (x^2 + 1)

((x^2 + 1)(5) – (5x – 2)(2x)) / (x^2 + 1)^2

(-5x^2 + 4x + 5) / (x^2 + 1)^2

d/dx[f(g(x))] = f’(g(x)) g’x

Example:y = (x^2 + 1)^3

dy/dx = 3(x^2 + 1)^2 (2x)

= 6x(x^2 + 1)^2

*This rule is very important*

Sin’(x) = cos(x)

Cos’(x) = -sin(x)

Tan’(x) = sec^2(x)

Cot’(x) = -csc^2(x)

Sec’(x) =sec(x)tan(x)

Csc’(x) = -csc(x)cot(x)

Examples: y = x – tan(x)y’ = 1 – sec(x)

y = x sec(x)y’ = x(secxtanx) +

(secx)(1)= (secx)(1+ x tanx)

d/dx[e^x] = e^x

Example: y = e^x^2 y’ = e^x^2 [d(x^2))/dx]y’ = 2x(e^x^2)

d/dx[ln(x)] = 1/x

Examples:h(x) = ln(5x) g(x) = ln(32x)h’(x) = 5(1/5x) g’(x) =

32(1/32x)h’(x) = 1/x g’(x) = 1/x