Calculate a Basic Derivative of a Function

This is intended as a guide to assist those who must occasionally calculate derivatives in generally non-mathematical courses such as economics, and can also be used as a guide for those just starting to learn calculus. This guide is meant for those who are already comfortable with algebra.

This guide is meant to provide one with the tools one will need to calculate derivatives of basic functions-- for an in-depth view of derivatives or for more advanced forms of differentiation such as the chain rule or partial differentiation, I recommend consulting the text Calculus: Early Transcendentals by James Stewart.

The symbol for a derivative that I will use in this guide is the ' symbol, I will use * for multiplication, and I will use ^ to indicate the use of an exponent.

Steps

Basic Overview of the Concept of a Derivative

A derivative is a calculation of the rate of change of a function. For instance, if you have a function that describes how fast a car is going from point A to point B, its derivative will tell you the car's acceleration from point A to point B-- how fast or slow the speed of the car changes. For more information on derivatives, see the note in "Calculating the Basic Derivative."

Simplify the function

  1. Know your algebra. Simplify the function at hand-- functions that are not simplified will still yield the same derivative, but it can be much more difficult to calculate.
    • Example:
      • Equation to simplify:
      • (6x + 8x)/2 +17x +4
      • Simplifying steps:
        • (14x)/2 + 17x + 4
        • 7x + 17x + 4
      • Final result:
        • 24x + 4

Identify the form of the function

  1. Learn the various forms.
    • Just a number (e.g. 4)
    • A number multiplied by a variable with no exponent (e.g. 4x)
    • A number multiplied by a variable with an exponent (e.g. 4x^2)
    • Addition (e.g 4x + 4)
    • Multiplication of variables (e.g. of the form x*x)
    • Division of variables (e.g. of the form x/x)

A number

  • The derivative of a function of this form is always zero.
    • Examples:
      • (4)' = 0
      • (-234059)' = 0
      • (pi)' = 0
        • Did you know? This is because there is no change in the function-- the value of the function will always be the number that you're given.

A number multiplied by a variable with no exponent

  • The derivative of a function of this form is always the number.
    • Examples:
      • (4x)' = 4
      • (x)' = 1
      • (-23x)' = -23
        • Did you know? If x does not have an exponent, the function is growing at a constant, steady, unchanging rate. You may recognize this trick from the linear equation y = mx + b.

A number multiplied by a variable with an exponent

  1. Multiply the number by the value of the exponent.
  2. Subtract one from the exponent.

Examples:

(4x^3)' = (4*3)(x^(3-1)) = 12x^2

(2x^7)' = 14x^6

(3x^(-1))' = -3x^(-2)



Addition

  1. Take the derivative of each part of the expression separately.

Examples:

(4x + 4)' = 4 + 0 = 4

((x^2) + 7x)' = 2x + 7

Multiplication of variables

1. Multiply the first variable by the derivative of the second variable.

2. Multiply the second variable by the derivative of the first variable.

3. Add your two results together.

Example:

((x^2)*x)' = (x^2)*1 + x*2x = (x^2) + 2x*x = 3x^2

Division of variables

1. Multiply the bottom variable by the derivative of the top variable.

2. Multiply the top variable by the derivative of the bottom variable.

3. Subtract your result in Step 2 from your result in Step 1. Be careful, order matters!

4. Divide your result in Step 3 with the square of the bottom variable.

Example:

((x^7)/x)' = (7x^6*x – 1*x^7)/(x^2) = (7x^7 - x^7)/(x^2) = 6x^7/x^2 = 6x^5

Warning: This is perhaps the hardest of the tricks to do, but it's well worth the effort. Make sure to do the steps in order and subtract in the correct order, and this will go smoothly.

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