Using the Quotient Rule in Calculus (With Step-by-Step Examples)

When functions show up as a fraction in calculus, the quotient rule is your go-to tool. Students often try to “wing it” with guessing or bad algebra, and that’s where marks disappear fast.

This guide explains the quotient rule in plain language, then walks through three fully worked examples (easy, medium, and hard). If you’re working with a tutor or studying on your own in Winnipeg, this is the kind of structure that keeps your derivative work clean and consistent.

What the Quotient Rule Says

Suppose you have a function written as a fraction:

y = f(x) / g(x),

where both f and g are differentiable and g(x) ≠ 0

The quotient rule says:

In words:

• Derivative of the top times the bottom,

• minus the top times the derivative of the bottom,

• all over the bottom squared.

When Should You Use the Quotient Rule?

Use the quotient rule when:

• You have one function divided by another, like those in the following image:

• Simplifying to a product first would be messy or introduce negative exponents students aren’t comfortable with yet.

  
  

The 4-Step Quotient Rule Recipe

For

y = f(x) / g(x)

you can use this routine:

1. Identify

   f(x) = numerator

   g(x) = denominator

2. Differentiate

   Find f'(x) and g'(x)

3. Apply the quotient rule

4. Simplify

   Expand where needed

   Factor if it makes the answer cleaner

Let’s put this into practice.

Example 1 (Easy): Polynomial over a Power of x

Differentiate

y = (3x² + 1) / x

Step 1: Identify f(x) and g(x)

• f(x) = 3x² + 1

• g(x) = x

Step 2: Differentiate each

• f'(x) = 6x

• g'(x) = 1

Step 3: Apply the quotient rule

Step 4: Simplify

Example 2 (Medium): Trig Function over a Power of x

Differentiate

y = sin(x) / x² , where x ≠ 0

Step 1: Identify f(x) and g(x)

• f(x) = sin(x)

• g(x) = x²

Step 2: Differentiate each

• f'(x) = cos(x)

• g'(x) = 2x

Step 3: Apply the quotient rule; you will get the following result.

Step 4: Simplify

That's the final, simplified answer.

Example 3 (Hard): Quotient with a Product Rule Inside

Now we combine two ideas: the quotient rule and the product rule.

Differentiate

y = (eˣ)(x² + 1) / (x² + 4)

We’ll treat the entire numerator as one function and the entire denominator as another.

Step 1: Identify f(x) and g(x)

• f(x) = (eˣ)(x² + 1)

• g(x) = (x² + 4)

Step 2: Differentiate each

• f'(x) = (eˣ)(x² + 1) + (eˣ)(2x) = (eˣ)(x² + 2x + 1) = (eˣ)(x + 1)²

• g'(x) = 2x

To differentiate f(x), we needed to use the product rule.

Click here to read my post on the product rule if you need a refresher.

After applying the product rule, we factored out the common factor eˣ and then factored the quadratic as a perfect square trinomial.

Step 3: Apply the quotient rule; you will get the following result.

Step 4: Simplify

Every term in the numerator has an eˣ, so factor it out.

You can expand further if your teacher expects it, but this is already a correct, nicely factored answer. If a student can reach this line accurately on a test, that’s strong work.

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Need help with the quotient rule? If you’re in Winnipeg and looking for a tutor, Tutor Advance provides expert one-on-one support!

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