Derivatives

The formal definition of a derivative is

lim h 0 f x + h - f x h

Stated simply, the derivative is an equation of a function that represents the slope of another line.

Here is an example of using the formal definition of the derivative to calculate the derivative of x to the second power.

  1. f x = x 2
  2. Start with the formal definition.

    f x = lim h 0 f x + h - f x h
  3. Put f(x) into the formal definition.

    f x = lim h 0 x + h 2 - x 2 h
  4. Simplify.

    f x = lim h 0 x 2 + 2 x h + h 2 - x 2 h
  5. Subtract x^2.

    f x = lim h 0 2 x h + h 2 h
  6. Cancel out h.

    f x = lim h 0 2 x + h
  7. Evaluate the limit.

    f x = 2 x

The final answer is 2x as you can see. Fortunately, the derivative can be found using a shortcut called the Power Rule.

Power Rule

The Power Rule is a shortcut for calculating the derivative of a term. Note: The notation d/dx will be used to signify "the derivative of". Here's the equation.

d dx x n = n x n - 1

Therefore, the derivative of 3x^2 would be

d dx 3 x 2 = 6 x

Derivative Table

Antiderivative Function Derivative - cos x sin x cos x sin x cos x - sin x - ln ( cos x ) tan x sec 2 x - ln | csc x + cot x | csc x - ( csc x ) ( cot x ) ln | sec x + tan x | sec x ( sec x ) ( tan x ) ln | sin x | cot x - csc 2 x