Derivative Rules (How To w/ 7+ Step-by-Step Examples!)

What are the derivative rules?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching the derivative rules

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Great question!

That’s exactly what we’re talking about in today’s calculus class, along with how to use them.

Here we go!

Let’s start by stating each of our differentiation rules in both words and symbols.

Power Rule

The power rule states that if n is any real number, then the derivative is:

How to find the instantaneous rate of change

General Version of the Power Rule

Sum and Difference Rule

The sum and difference rule for derivatives states that if f(x) and g(x) are both differentiable functions, then:

Derivative Sum Difference — Formula

Constant Multiple Rule

The constant multiple rule states that if c is a constant and f(x) is a differentiable function, then:

Constant Multiple Rule

Product Rule

The product rule states that if f(x) and g(x) are both differentiable functions, then the derivative of their product also exists.

How to differentiate a product of two functions

The Product Rule Explained

Quotient Rule

The quotient rule states that if f(x) and g(x) are both differentiable functions and g(x) does
not equal zero, then the derivative of their quotient also exists.

Quotient Rule Formula

Chain Rule

The chain rule states that if f(x) and g(x) are both differentiable functions and the composite function defined as F=f(g(x)), then F is differentiable, and F’ is given by the product.

Calculating the derivative of composite functions

Understanding the Chain Rule

Example – Combinations

As we will quickly see, each derivative rule is necessary and useful for finding the instantaneous rate of change of various functions.

More importantly, we will learn how to combine these differentiations for more complex functions.

For example, suppose we wish to find the derivative of the function shown below.

Find The Derivative Of The Function

What do we notice?

Not only do we have a quotient rule, but also a chain rule as well!

Therefore, this is what I like to call a combination question, or “combo,” as we will need to employ two differentiation techniques to complete our goal.

Combining Derivative Rules

Overall, these types of questions aren’t too bad, as long as we stop to think through what our steps should be and take our time as we simplify.

So, throughout this lesson, we will not only refine and check our skills in calculating derivatives, we will tackle questions for when two or more differentiation rules are needed.

Alright, let’s jump right in.

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

calcworkshop jenn teaching derivative rules

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