Homework Help: Calculus Rules for Derivatives

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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)

This post is part of the series: Calculus Help

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  1. What Are the Calculus Derivative Rules?