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What Are the Calculus Derivative Rules?

By Kayla Griffin

Learn here everything from the rules (power rule, constant function and chain rule) to exponential and logarithmic functions. You can look at all the rules for any derivative you need.

Notes on Notation

For the student who needs help with calculus, this is a reference to help provide easy access to derivative rules. Be certain to bookmark this page so that you can return to it again as needed.

Assume that u, v, f, g represent functions: u(x), v(x), f(x), g(x)

Assume that k, a are constants.

Assume that x, and y are variables.

The letter e is used to represent the exponential function.

The logarithmic functions are denoted: ln (natural log or loge), and loga

f'(x) means the derivative of f with respect to x, or df/dx.

If f(x) is a function, then f-1(x) is its inverse function.

Basic Rules

Derivative of x:

If f(x) = x, then f'(x) = 1

Sum Rule:

The derivative of the sum or difference of two functions is the sum or difference of the derivatives

if f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x)

Constant Multiple Rule:

Remember that the derivative of any constant will be zero. You can be confident that constants will not change with any variable.

The derivative of a function multiplied by a constant is the derivative of the function multiplied by the same constant.

If f(x) = k*u(x), then f'(x) = k*u’(x)

Constant Function Rule:

The derivative of a constant is zero.

f(x) = k; f’(x) = 0

Power Rule:

If f(x) = xn , then f’(x) = n*x(n-1)

Product Rule:

The derivative of a product is the first factor times the derivative of the second factor plus the second factor times the derivative of the first factor.

If f(x) = u(x)*v(x), then f'(x) = u(x)v’(x)-v(x)u’(x)

Quotient Rule:

The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared

If f(x) = u(x)/v(x), then f'(x) = [v(x)u’(x)-u(x)v’(x)] / [v(x)]2

Chain Rule:

The Chain rule is used for composite functions: e.g. f(g(x)). The derivative of a composition u(v(x)) is the derivative of u evaluated at v(x) multiplied by the derivative of vi(x)

f(x) = u(v(x)); f’(x) = u’(v(x))*v’(x)

Derivative of Absolute Value:

If f(x) = |u(x)|, then f'(x) = [u(x)/|u(x)|]*u(x)

Derivative of an Inverse Function:

Let f be a function that is differentiable on an interval [a,b]. If f has an inverse function g, then g is differentiable at any x for which f’(g(x)) is not 0.

If g(x) = f-1(x), then g’(x) = 1/(f’g(x)), f’(g(x)) ≠ 0

Logarithmic and Exponential Functions

Natural Log (ln)

Assume u(x) is differentiable.

If f(x) = ln(x), then f’(x) = 1/x

If f(x) = ln(u(x)), then f’(x) = u’(x)/u(x)

Exponential Function:

F(x) = e^x; f’(x) = e^x

F(x) = e^u(x); f’(x) = e^u(x)*u’(x)

Derivatives for bases other than e

Let a be a positive real number greater than 1. And let u be a differentiable function of x

If f(x) = ax then f’(x) = ln(a)*ax

If f(x) = au(x), then f’(x) = (ln(a))*au(x)*u’(x)

If f(x) = loga(x), then f’(x) = 1/(ln(a)*x)

If f(x) = logau(x), then f’(x) =[ 1/(ln(a)*u(x))]*u’(x)