Skip to main content

Derivative Rules

In calculus, a derivative can be thought of as an instantaneous rate of change; that is, how much a quantity is changing at a given point. Let’s take a closer look at how we can differentiate a function easily by the use of some helpful rules.

Understanding the Derivative

Differentiation is a method to compute the rate at which a dependent variable y changes with respect to the change in the independent variable x. This rate of change is called the derivative of y with respect to x. There are many different notations to denote “take the derivative of.” The relationship between y and x is usually denoted by f(x) and its derivative is usually denoted as f‘(x) or y’ or dy/dx. The definition of the derivative is given by:

 derivatives equation

which is a lengthy procedure used to evaluate the derivative of a function. Thankfully, easier methods have been developed to help evaluate derivatives more quickly.

The Power Rule

If n is any real number, then  d (xn) = nxn-1
                                                dx                                                                                                                   
Also, remember that the derivatives of a constant is zero:  (c) = 0.
                                                                                                dx

Note that the sum and difference rule states:

 The power rule

(Just simply apply the power rule to each term in the function separately).

Example: Find the derivative of

derivative example

Solution:

First, rewrite the function so that all variables of x have an exponent in the numerator:

derivative example

Now, apply the power rule to the function:

derivative example

Lastly, simplify your derivative:

derivative example

The Product Rule

We already know how to find the derivative of a sum or difference of functions, but what about the product of two functions? The product of two functions is when two functions are being multiplied together. If f and g are both differentiable, then the product rule states: 

The Product Rule equation

Example: Find the derivative of h(x) = (3x + 1)(8x4 +5x).

Solution:

Using the above formula, let f(x) = (3x+1) and let g(x) = (8x4 + 5x).

Now, find 

product rule example

Lastly, apply the product rule using the above formula:

product rule example

The Quotient Rule

Now, let's take a look at the quotient of functions (when two functions are being divided by each other). If f and g are both differentiable, then the quotient rule states:

quotient rule

Example: Find the derivative of 

quotient rule example

Solution:

Using the above formule, let f(x) = (3x-4) and let g(x) = (2x2-1).

Now, find

quotient rule example

Lastly, apply the quotient rule using the above formula:

quotient rule example

The Chain Rule

If f and g are both differentiable and F is the composite function defined by F(x) = f(g(x)), then F is differentiable and the chain rule can be applied to F ', which is given by  F '(x) = f '(g(x))g '(x)

Example: Find the derivative of F(x) = (3x4 + 2x2 -7)5.

Solution:

Using the above formula, let g(x) = (3x4 + 2x- 7) and let f(x) = (x)5.

Now, find f '(x) and '(x):  f '(x) = 5x4 and '(x) = 12x3 + 4x - 0

So, if f '(x) = 5x4, then the composite function f '(g(x)) = 5(3x4 + 2x2 - 7)4

Lastly, apply the chain rule using the above formula: '(x) = f '(g(x))g '(x)

'(x)  = 5(3x4 + 2x2 -7)4 (12x3 + 4x)

University of Ontario Institute of Technology logo