Derivative rules 408D

Product Rule

The rule for differentiating the product of two functions: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

Quotient Rule

The rule for differentiating the quotient of two functions: d/dx [f(x)/g(x)] = (g(x)f'(x) - f(x)g'(x)) / [g(x)]^2.

Chain Rule

The rule for differentiating composite functions: d/dx [f(g(x))] = f'(g(x))g'(x).

d/dx [e^x]

The derivative of e raised to the power of x is e^x.

Special Case of Chain Rule

For d/dx [e^(f(x))], the derivative is e^(f(x))f'(x).

d/dx [ln x]

The derivative of the natural logarithm of x is 1/x.

Special Case of Chain Rule

For d/dx [ln(f(x))], the derivative is f'(x)/f(x).

d/dx [a^x]

The derivative of a raised to the power of x is a^x ln a.

d/dx [sin x]

The derivative of sine of x is cos x.

d/dx [cos x]

The derivative of cosine of x is -sin x.

d/dx [tan x]

The derivative of tangent of x is sec^2 x.

d/dx [sec x]

The derivative of secant of x is sec x tan x.

d/dx [csc x]

The derivative of cosecant of x is -csc x cot x.

d/dx [cot x]

The derivative of cotangent of x is -csc^2 x.

d/dx [sin^(-1) x]

The derivative of the inverse sine of x is 1/√(1-x^2).

d/dx [cos^(-1) x]

The derivative of the inverse cosine of x is -1/√(1-x^2).

d/dx [tan^(-1) x]

The derivative of the inverse tangent of x is 1/(1+x^2).

d/dx [sec^(-1) x]

The derivative of the inverse secant of x is 1/|x|√(x^2-1).

d/dx [csc^(-1) x]

The derivative of the inverse cosecant of x is -1/|x|√(x^2-1).

d/dx [cot^(-1) x]

The derivative of the inverse cotangent of x is -1/(1+x^2).