Homework Help: Calculus Rules for Derivatives
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
Function lists and cheat sheets, step-by-step instructions and examples, everything you need to pass your calculus class can be found here.
