Product Rule

The product rule is not simply that the derivative of multiplied functions is the product of their separate derivatives. Because both functions change at the same time, the rule accounts for this by differentiating one factor while keeping the other unchanged, then adding the results together.

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more differentiable functions, then the product is differentiable. For two functions, it may be stated in Lagrange’s notation as

(u ⋅ v)’ = u’ ⋅ v + u ⋅ v’

or in Leibniz’s notation as

$\frac{d}{dx}\left( u\cdot v \right) \,\,=\,\,\frac{du}{dx}\cdot v\,\,+\,\,u\cdot \frac{dv}{dx}$

Product Rule Proof

Rather than detail the proof here, see the proof in the Product Rule Proof section on the What Is the Product Rule? [With Examples] webpage. Knowing the proof will help you understand how the product rule works and why it can be used for differentiable functions.

Note the following as you read the proof.

The product rule not just a formula to memorize. It’s what happens when both parts of a product are changing at the same time.
Pay attention to the step where f(x + Δx)g(x) – f(x + Δx)g(x) is added to the numerator (algebraic manipulation), essentially adding zero, to make the equation easier to solve.
Note the use of the sum law for limits to separate the expression into a sum of limits, and the use of the product law for limits.

Geometric Illustration

Imagine the area of a rectangle with sides of width u ( f(x) ) and height v ( g(x) ). When x changes a tiny bit (Δx), both sides change (Δu and Δv). The total change in area comes from:

The change due to u changing (while v stays roughly the same)
The change due to v changing (while u stays roughly the same)

Step 1: Geometric Setup

Let h(x) = f(x) ⋅ g(x)

When x increases by a small amount Δx > 0, the width changes by Δf = f(x + Δx) – f(x), and the height changes by Δg = g(x + Δx) – g(x).

Step 2: Expand the New Area Algebraically

The new area is: [ f(x) + Δf ] ⋅ [ g(x) + Δg ]

Multiply it out: [ f(x) + Δf ] ⋅ [ g(x) + Δg ] = f ⋅ g + f ⋅ Δg + Δf ⋅ g + Δf ⋅ Δg = fg ⋅ fΔg + gΔf + Δf ⋅ Δg

The change in area ΔA = h(x + Δx) – h(x)

is therefore

ΔA = fΔg + gΔf + Δf ⋅ Δg (or uΔv + vΔu + ΔuΔv),

We can therefore say that the sum of the blue areas and the gray area in the geometric picture is the derivative of f(x)g(x).

This matches the geometric picture:

fΔg ⇾ the horizontal strip (full original width (u) x height change (Δv))
gΔf ⇾ the vertical strip (full original height (v) x width change (Δu))
Δf ⋅ Δg ⇾ small corner rectangle (ΔuΔv)

*Geometric illustration of a proof of the product rule – Wikipedia*

Step 3: Rate of Change (Divide by Δx)

The average rate of change of the area is:

$\frac{\varDelta A}{\varDelta x}\,=\,\frac{f\varDelta g\,+\,g\varDelta f\,+\,\varDelta f \cdot \varDelta g}{\varDelta x}$

Split into three terms:

$\frac{\varDelta A}{\varDelta x}\,=\,f\left( x \right) \cdot \frac{\varDelta g}{\varDelta x}\,+\,g\left( x \right) \cdot \frac{\varDelta f}{\varDelta x}\,+\,\frac{\varDelta f \cdot \varDelta g}{\varDelta x}$

Step 4: Take the Limit as Δx ⇾ 0

The derivative h’(x) is:

$h\prime\left( x \right) \,\,=\,\,\lim_{\varDelta \,\,x\,\,\rightarrow 0} \,\,\frac{\varDelta \,\,A}{\varDelta \,\,x}\,\,=\,\,\lim_{\varDelta \,\,x\,\,\rightarrow 0} \,\,\left[ f\left( x \right) \,\,\cdot \,\,\frac{\varDelta \,\,g}{\varDelta \,\,x}\,\,+\,\,g\left( x \right) \,\,\cdot \,\,\frac{\varDelta \,\,f}{\varDelta \,\,x}\,\,+\,\,\frac{\varDelta \,\,f\,\,\cdot \varDelta \,\,g}{\varDelta \,\,x} \right]$

Next we evaluate each term.

First Term

f(x) does not depend on Δx, so it stays outside.

$\lim_{\varDelta \,\,x\,\,\rightarrow 0} \,\,\frac{\varDelta \,\,g}{\varDelta \,\,x}\,\,=\,\,g\prime\left( x \right) \,\,by\,\,definition$

⇾ f(x)g‘(x)

Second Term

Similarly, g(x) is constant with respect to Δx.

$\lim_{\varDelta \,\,x\,\,\rightarrow 0} \,\,\frac{\varDelta \,\,f}{\varDelta \,\,x}\,\,=\,\,f\prime\left( x \right)$

⇾ g(x)f‘(x)

Third Term (the corner)

Rewrite as:

$\frac{\varDelta \,\,f\,\,\cdot \,\,\varDelta \,\,g}{\varDelta \,\,x}\,\,=\,\,\varDelta \,\,f\,\,\cdot \,\,\left( \frac{\varDelta \,\,g}{\varDelta \,\,x} \right)$

As Δx ⇾ 0:

Δg/Δx approaches the finite value g‘(x) (assuming g’ exists).
Δf ⇾ 0 (because differentiability implies continuity)

Therefore, the product

$\varDelta \,\,f\,\,\cdot \,\,\left( \frac{\varDelta \,\,g}{\varDelta \,\,x} \right) \,\, \rightarrow \,\,0 \cdot \,\,g\prime\left( x \right) \,\,=\,\,0$

Step 5: Final Result

Putting it all together:

$h\prime\left( x \right) \,\,\\=\,\,f\left( x \right) g\prime\left( x \right) \,\,+\,\,g\left( x \right) f\prime\left( x \right) \,\,+\,\,0 \\=\,\,f\prime\left( x \right) g\left( x \right) \,\,+\,\,f\left( x \right) g\prime\left( x \right)$

This is the product rule.

Key Algebraic Insight

This geometric view was used intuitively by Leibniz with infinitesimals. It gives strong intuition for why the product rule has two main terms: one for each changing side plus a vanishing cross term. The corner term must vanishes because it is a higher-order infinitesimal (roughly proportional to Δx²) while the other two terms are order Δx. When divided by Δx, the first two become finite, and the corner term goes to zero.

References

ALLEN. “Continuity and Differentiability.” ALLEN, August 13, 2024. https://allen.in/jee/maths/continuity-and-differentiability.

Cazoom Maths. “Product Rule — Definition, Formula & Examples.” Mathwords, 2026. https://www.mathwords.com/p/product_rule.htm.

“Continuity and Discontinuity.” BYJU’S, November 6, 2019. https://byjus.com/maths/continuity-and-discontinuity/.

“If f is continuous at x = c, then f is differentiable at x = c ? [SOLVED].” Accessed June 9, 2026. https://www.cuemath.com/questions/if-f-is-continuous-at-x-c-then-f-is-differentiable-at-x-c/.

Product Rule

Dawkins, Paul. “Calculus I.” Product and Quotient Rule, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/productquotientrule.aspx.

[ ℰ ] McLean, Rachel. “What Is the Product Rule? [With Examples].” Outlier, July 25, 2022. https://articles.outlier.org/what-is-the-product-rule.

[ ℰ ] Nykamp, Duane. “The Idea of the Product Rule.” Math Insight, 2020. https://mathinsight.org/product_rule_idea.

“Product-Rule.” BYJU’S, October 10, 2019. https://byjus.com/maths/product-rule/.

“Product Rule.” CalcWorkshop. September 6, 2015. https://calcworkshop.com/derivatives/product-rule/.

“Product Rule.” Wikipedia, February 6, 2026. https://en.wikipedia.org/wiki/Product_rule.

[ ℰ ] “Product Rule Explained Step by Step with Examples – NUM8ERS.” NUM8ERS. Accessed June 10, 2026. https://num8ers.com/guides/product-rule/.

[ ℰ ] Sanderson, Lesson by Grant, and Text adaptation by Kurt Bruns. “Visualizing the Chain Rule and Product Rule.” 3Blue1Brown, May 1, 2017. https://www.3blue1brown.com/lessons/chain-rule-and-product-rule/.

Sum Law for Limits

Dawkins, Paul. “Calculus I.” Proof of Various Limit Properties, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/limitproofs.aspx.

Strang, Gilbert, and Edwin “Jed” Herman. “2.3 The Limit Laws – Calculus Volume 1.” OpenStax, March 30, 2016. https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws.

The Department of Mathematics and Computer Science. “Proof of the Limit of a Sum Law.” Emory Oxford College. Accessed June 7, 2026. https://mathcenter.oxford.emory.edu/site/math111/proofs/limitOfSum/.

UT Calculus. “Limit Laws and Computations.” web.ma.utexas.edu. Accessed June 7, 2026. https://web.ma.utexas.edu/users/m408n/AS/LM2-3-2.html.

Product Law for Limits

Dawkins, Paul. “Calculus I.” Proof of Various Limit Properties, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/limitproofs.aspx.

Strang, Gilbert, and Edwin “Jed” Herman. “2.3 The Limit Laws – Calculus Volume 1.” OpenStax, March 30, 2016. https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws.

UT Calculus. “Limit Laws and Computations.” web.ma.utexas.edu. Accessed June 7, 2026. https://web.ma.utexas.edu/users/m408n/AS/LM2-3-2.html.

Videos

Product Rule For Derivatives

@TehraByte. “The Product Rule for Derivatives: Simple Examples to Understand This Key Calculus Concept.” YouTube. 2008. Video. https://www.youtube.com/watch?v=uPCjqfT0Ixg.

In this video, we’ll go over the product rule formula for finding derivatives. I’ll walk you through three examples to help solidify your understanding of applying the product rule. The first two examples cover the basics, and in the final example, we’ll tackle a more complex case involving three factors—requiring us to use the product rule within the product rule.

What You Will Learn:
The formula for the product rule in calculus.
How to apply the product rule to find derivatives of functions with multiple factors.
Step-by-step examples, from straightforward to more complex cases.
How to handle derivatives when three or more factors are involved.
Tips for simplifying and correctly applying the product rule in various scenarios.

Proof of Sum Law (Limit Laws)

Limit Laws – Proof of Product Law

3.2.4 Differentiability Implies Continuity

Notes

Algebraic Manipulation

Algebraic manipulation is the process of rearranging, simplifying, or transforming algebraic expressions and equations into a more useful form. The goal is usually to isolate a specific variable, simplify a complex formula, or solve for an unknown value while ensuring the fundamental truth of the equation remains unchanged.

Differentiability

Differentiability is a key concept in calculus that indicates whether a function has a derivative at a given point. A function is differentiable at a point if it is continuous there and its derivative exists. This property is crucial when applying the Product Rule, as it ensures that the functions involved can be differentiated, allowing for the application of the rule to find the derivative of their product.

Limit of a Sum Equal to the Sum of the Limits

The limit of a sum is equal to the sum of the limits because of how limits are defined. A limit describes what happens when numbers get closer to a target number. Because we can make two separate numbers as close to their targets as we want, their combined sum gets as close to the combined targets as we want.

The Core Concept

The limit: The exact value two parts “approach”.
The rule: lim [f(x) + g(x)] = lim f(x) + lim g(x).
The condition: This rule only works if the individual limits exist and are finite numbers.

The Grocery Store Analogy

Imagine you are filling two bags of groceries.

Bag A will eventually weigh exactly 10 pounds.
Bag B will eventually weigh exactly 15 pounds.
If you ask how much the total groceries weigh in the end, you add the limits: 10 + 15 = 25 pounds.
The total limit is the sum of the individual limits.

Why the Rule Doesn’t Always Work

While the rule works for finite steps, it fails for infinite sums unless strict conditions are met. [1, 2]

Infinite terms: Adding an infinite number of terms sometimes requires a special property called uniform convergence.
Without this property, adding the individual limits can give a different answer than doing the infinite sum first.
A famous example is Gabriel’s Horn, where you can have an infinite surface area but a finite volume when you do the limit correctly.

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