Clear, step‑by‑step examples of product and quotient rules in derivatives

Starting with simple product rule examples

Let’s jump straight into the product rule with concrete problems. The product rule is used when you’re differentiating a product of two functions, like \(f(x) = u(x) \cdot v(x)\). The rule says:

[
(fg)’(x) = f’(x)g(x) + f(x)g’(x).
]

Think of it as: derivative of the first times the second, plus the first times the derivative of the second.

Example of a basic algebraic product

Suppose
[
f(x) = (3x^2)(5x^3).
]
You could multiply this out first to get \(f(x) = 15x^5\), then differentiate easily. But let’s treat it as a product to practice the rule.

Let \(f(x) = u(x)v(x)\) with

• \(u(x) = 3x^2\), so \(u’(x) = 6x\)
• \(v(x) = 5x^3\), so \(v’(x) = 15x^2\).

Apply the product rule:
[
\begin{aligned}
f’(x) &= u’(x)v(x) + u(x)v’(x) \
&= (6x)(5x^3) + (3x^2)(15x^2) \
&= 30x^4 + 45x^4 \
&= 75x^4.
\end{aligned}
]

If you had multiplied first, \(15x^5\) differentiates to \(75x^4\), which matches. This is one of the best examples for seeing that the product rule and “expand then differentiate” can agree when that expansion is easy.

Trig and polynomial: a classic product rule example

Now let’s look at a more realistic homework‑style problem:
[
f(x) = x^2 \sin x.
]
This is a product of a polynomial and a trigonometric function.

Let

• \(u(x) = x^2\), so \(u’(x) = 2x\)
• \(v(x) = \sin x\), so \(v’(x) = \cos x\).

Apply the rule:
[
\begin{aligned}
f’(x) &= u’(x)v(x) + u(x)v’(x) \
&= (2x)(\sin x) + (x^2)(\cos x) \
&= 2x\sin x + x^2\cos x.
\end{aligned}
]

This is one of those real examples that shows up again and again in calculus courses and AP practice problems.

Exponential and trig: a slightly spicier product

Consider
[
f(x) = e^x \cos x.
]
This is a favorite in many modern online problem sets. It’s also a nice example of how the product rule combines with familiar derivatives.

Let

• \(u(x) = e^x\), so \(u’(x) = e^x\)
• \(v(x) = \cos x\), so \(v’(x) = -\sin x\).

Then
[
\begin{aligned}
f’(x) &= u’(x)v(x) + u(x)v’(x) \
&= e^x \cos x + e^x(-\sin x) \
&= e^x(\cos x - \sin x).
\end{aligned}
]

You’ll see this kind of pattern in differential equations, signal processing, and even in some physics‑style oscillation models.

Real examples of product rule in applied contexts

To make these examples of product and quotient rules in derivatives feel less abstract, let’s frame the product rule in a real‑style context.

Example: Revenue as price × quantity

Imagine a simple economics‑style setup where revenue \(R(t)\) depends on time \(t\):
[
R(t) = p(t) \cdot q(t),
]
where

• \(p(t)\) is price in dollars
• \(q(t)\) is quantity sold per day.

If both price and quantity change over time (think 2024–2025 online sales that respond to demand and promotions), then the rate of change of revenue is
[
R’(t) = p’(t)q(t) + p(t)q’(t).
]

In words: change in revenue comes from two sources:

• price changing while quantity is held at its current level
• quantity changing while price is held at its current level.

This is one of the best examples of product rule thinking: each factor changes, and the rule neatly captures both effects.

For a quick numeric version, suppose
[
p(t) = 20 + t, \quad q(t) = 100 - 2t.
]
Then
[
p’(t) = 1, \quad q’(t) = -2,
]
so
[
R’(t) = (1)(100 - 2t) + (20 + t)(-2) = 100 - 2t - 40 - 2t = 60 - 4t.
]

Here you’ve used the product rule to understand how revenue changes day by day.

Moving to quotient rule examples

Now let’s switch to the quotient rule, which you use when your function is a ratio:
[
f(x) = \frac{u(x)}{v(x)}.
]

The quotient rule says:
[
\left(\frac{u}{v}\right)’ = \frac{u’v - uv’}{v^2}, \quad v(x) \ne 0.
]

A good way to remember it: bottom times derivative of top, minus top times derivative of bottom, all over bottom squared. Order matters in that subtraction.

Example of a rational function

Take
[
f(x) = \frac{x^2 + 1}{x^3}.
]
You could rewrite this as \(x^{-1} + x^{-3}\) and differentiate term by term, but let’s treat it as a quotient to practice.

Let

• \(u(x) = x^2 + 1\), so \(u’(x) = 2x\)
• \(v(x) = x^3\), so \(v’(x) = 3x^2\).

Apply the quotient rule:
[
\begin{aligned}
f’(x) &= \frac{u’v - uv’}{v^2} \
&= \frac{(2x)(x^3) - (x^2 + 1)(3x^2)}{(x^3)^2} \
&= \frac{2x^4 - 3x^4 - 3x^2}{x^6} \
&= \frac{-x^4 - 3x^2}{x^6} \
&= -\frac{1}{x^2} - \frac{3}{x^4}.
\end{aligned}
]

If you’d rewritten first as \(x^{-1} + x^{-3}\), you’d get the same result. This is a clean example of how the quotient rule and algebraic simplification can work together.

Trig over trig: a classic quotient rule example

Consider
[
f(x) = \frac{\sin x}{x}.
]
This function shows up in signal processing and Fourier analysis, and it’s one of the most common real examples of quotient rule practice.

Let

• \(u(x) = \sin x\), so \(u’(x) = \cos x\)
• \(v(x) = x\), so \(v’(x) = 1\).

Then
[
\begin{aligned}
f’(x) &= \frac{u’v - uv’}{v^2} \
&= \frac{(\cos x)(x) - (\sin x)(1)}{x^2} \
&= \frac{x\cos x - \sin x}{x^2}.
\end{aligned}
]

You’ll see this exact derivative in many calculus textbooks and open‑course materials from universities like MIT and Harvard.

Mixed examples of product and quotient rules in derivatives

Now let’s look at functions where you’ll need both rules. These mixed cases give some of the best examples of product and quotient rules in derivatives working together.

Example: Product of a power and a quotient

Take
[
f(x) = x^2 \cdot \frac{\ln x}{x+1}.
]
This is a product of \(x^2\) and a quotient. You’ll use the product rule first, and inside that, the quotient rule.

Let

• \(u(x) = x^2\), so \(u’(x) = 2x\)
• \(v(x) = \dfrac{\ln x}{x+1}\).

First, find \(v’(x)\) using the quotient rule.

For \(v(x)\), set

• numerator: \(a(x) = \ln x\), so \(a’(x) = 1/x\)
• denominator: \(b(x) = x+1\), so \(b’(x) = 1\).

Then
[
\begin{aligned}
v’(x) &= \frac{a’b - ab’}{b^2} \
&= \frac{(1/x)(x+1) - (\ln x)(1)}{(x+1)^2} \
&= \frac{(x+1)/x - \ln x}{(x+1)^2}.
\end{aligned}
]

Now apply the product rule to \(f(x) = u(x)v(x)\):
[
\begin{aligned}
f’(x) &= u’(x)v(x) + u(x)v’(x) \
&= 2x \cdot \frac{\ln x}{x+1} + x^2 \cdot \frac{(x+1)/x - \ln x}{(x+1)^2}.
\end{aligned}
]

You can simplify further if needed, but the main idea is seeing how examples of product and quotient rules in derivatives can be layered.

Example: Quotient of a product and a function

Now try
[
f(x) = \frac{x e^x}{x^2 + 1}.
]
Here the numerator is a product, and the whole thing is a quotient.

Let

• \(u(x) = x e^x\)
• \(v(x) = x^2 + 1\).

First find \(u’(x)\) using the product rule.

For \(u(x) = x e^x\):

• \(a(x) = x\), \(a’(x) = 1\)
• \(b(x) = e^x\), \(b’(x) = e^x\).

Then
[
\begin{aligned}
u’(x) &= a’(x)b(x) + a(x)b’(x) \
&= 1\cdot e^x + x\cdot e^x \
&= e^x(1 + x).
\end{aligned}
]

For the denominator, \(v(x) = x^2 + 1\), we have \(v’(x) = 2x\).

Now apply the quotient rule to \(f(x) = u(x)/v(x)\):
[
\begin{aligned}
f’(x) &= \frac{u’v - uv’}{v^2} \
&= \frac{e^x(1 + x)(x^2 + 1) - (x e^x)(2x)}{(x^2 + 1)^2} \
&= \frac{e^x\big[(1 + x)(x^2 + 1) - 2x^2\big]}{(x^2 + 1)^2}.
\end{aligned}
]

This kind of expression shows up in growth and decay models where an exponential term is moderated by a rational factor.

Logarithmic and exponential combinations

Because 2024–2025 calculus courses and online platforms lean heavily on exponential and logarithmic functions, let’s look at more real examples involving these.

Example: Product of log and polynomial

Consider
[
f(x) = (x^3 - 1)\ln x.
]
This is a straightforward product rule problem.

Let

• \(u(x) = x^3 - 1\), so \(u’(x) = 3x^2\)
• \(v(x) = \ln x\), so \(v’(x) = 1/x\).

Then
[
\begin{aligned}
f’(x) &= u’(x)v(x) + u(x)v’(x) \
&= 3x^2 \ln x + (x^3 - 1)\cdot\frac{1}{x} \
&= 3x^2 \ln x + x^2 - \frac{1}{x}.
\end{aligned}
]

This is one of the best examples to practice combining the product rule with the derivative of \(\ln x\).

Example: Quotient of exponential and polynomial

Take
[
f(x) = \frac{e^{2x}}{x^2 + 4}.
]

Let

• \(u(x) = e^{2x}\), so \(u’(x) = 2e^{2x}\)
• \(v(x) = x^2 + 4\), so \(v’(x) = 2x\).

Apply the quotient rule:
[
\begin{aligned}
f’(x) &= \frac{u’v - uv’}{v^2} \
&= \frac{2e^{2x}(x^2 + 4) - e^{2x}(2x)}{(x^2 + 4)^2} \
&= \frac{e^{2x}\big[2(x^2 + 4) - 2x\big]}{(x^2 + 4)^2} \
&= \frac{2e^{2x}(x^2 - x + 4)}{(x^2 + 4)^2}.
\end{aligned}
]

You’ll see expressions like this in probability density functions and statistics, where exponentials and rational terms often appear together.

When to use product vs. quotient rule (and when not to)

A lot of students in 2024–2025 are learning from video platforms and apps that throw mixed problems at them. It helps to recognize patterns:

• Use the product rule when you have a clear multiplication of two differentiable functions, and expanding would be messy or impossible (like trig times exponential).
• Use the quotient rule when the function is a clean ratio and rewriting as a product with negative exponents would be more confusing.
• Sometimes, you can simplify first to avoid the quotient rule. For example, \(\dfrac{x^3}{x^2} = x\), which needs only the power rule.

These real examples of product and quotient rules in derivatives are not about memorizing tricks; they’re about seeing structure:

• Is this a product?
• Is this a quotient?
• Can I rewrite it to make life easier?

For more background on differentiation rules and practice problems, you can check open resources like MIT OpenCourseWare, Harvard’s math course materials, or general calculus overviews from Khan Academy (a .org nonprofit).

FAQ: common questions about product and quotient rule examples

What are some easy examples of product rule in derivatives?

Easy examples include functions like \(f(x) = x^2\sin x\), \(f(x) = x e^x\), or \(f(x) = (3x^2)(5x^3)\). Each is a clear product of two functions you already know how to differentiate. These are often the first examples of product rule that appear in high school and early college calculus.

Can you give an example of when both product and quotient rules are needed?

Yes. A good example of this is
[
f(x) = x^2 \cdot \frac{\ln x}{x+1}.
]
You treat it as a product of \(x^2\) and a quotient. First, you use the quotient rule to differentiate \(\dfrac{\ln x}{x+1}\), then plug that into the product rule. This is one of the best examples of product and quotient rules in derivatives working together in a single problem.

Are there real examples where these rules show up outside of math class?

Absolutely. In physics‑style problems, you might see position as \(s(t) = t e^{-t}\), a product of time and an exponential decay, so velocity uses the product rule. In economics‑style models, revenue \(R(t) = p(t)q(t)\) uses the product rule. In probability and statistics, ratios like \(\dfrac{e^{-x^2}}{1+x^2}\) require the quotient rule. These are all realistic examples of product and quotient rules in derivatives applied to modeling.

How do I know if I should simplify before using the quotient rule?

Look for common factors or obvious cancellations. For instance, \(\dfrac{x^3}{x^2 + x}\) can be simplified by factoring the denominator: \(x^2 + x = x(x+1)\). Then
[
\frac{x^3}{x(x+1)} = \frac{x^2}{x+1}.
]
Now the derivatives are easier. Many instructors and resources, including university calculus notes (for example, materials from UT Austin’s math department), encourage simplifying first when it clearly reduces the algebra.

Where can I find more practice problems and examples of product and quotient rules in derivatives?

You can find more worked examples and practice sets from:

• University course pages, such as MIT OpenCourseWare’s single variable calculus
• Nonprofit learning platforms like Khan Academy’s calculus section
• Open textbooks hosted by universities and organizations like OpenStax.

Working through many examples of examples of product and quotient rules in derivatives is the fastest way to turn these rules from something you have to think about into something that feels almost automatic.