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Product Rule Calculator

Apply the product rule to differentiate the product of two power functions. Enter coefficients and exponents of two terms f(x) = ax^m and g(x) = bx^n to find d/dx[f*g], essential for calculus students working with products of functions.

Reviewed by Christopher FloiedPublished Updated

This free online product rule calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient a in f(x) = ax^m

Exponent m in f(x) = ax^m

Coefficient b in g(x) = bx^n

Exponent n in g(x) = bx^n

Results

Combined Coefficient

50

Combined Exponent

4

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Product Rule Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Product Rule Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Product Rule Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Product Rule Calculator

The Product Rule Calculator computes the derivative of a product of two power functions f(x) = ax^m and g(x) = bx^n. The product rule states that d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x). For the special case of two power functions, the product simplifies to a single power term ab(m+n)x^(m+n-1). This calculator demonstrates both the rule and its simplification, making it an excellent tool for verifying homework, understanding how the product rule works, and building intuition for more complex products involving trigonometric, exponential, or logarithmic functions. The product rule is indispensable in physics, engineering, and economics.

The Math Behind It

The product rule is a fundamental differentiation formula: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). The proof follows directly from the definition of the derivative. Consider the limit [f(x+h)g(x+h) - f(x)g(x)]/h. By adding and subtracting f(x+h)g(x) in the numerator, this becomes f(x+h)[g(x+h)-g(x)]/h + g(x)[f(x+h)-f(x)]/h. As h approaches 0, this gives f(x)g'(x) + g(x)f'(x). For two power functions f = ax^m and g = bx^n, applying the product rule: f'g + fg' = (amx^(m-1))(bx^n) + (ax^m)(bnx^(n-1)) = abmx^(m+n-1) + abnx^(m+n-1) = ab(m+n)x^(m+n-1). This is the same result you get by first multiplying to get abx^(m+n) and differentiating directly, confirming the product rule's correctness. The product rule generalizes to three or more functions: d/dx[fgh] = f'gh + fg'h + fgh'. Each factor gets differentiated exactly once while the others remain unchanged. Leibniz actually discovered a pattern analogous to the binomial theorem for the nth derivative of a product, known as Leibniz's rule: (fg)^(n) = sum over k of (n choose k) f^(k) g^(n-k). Applications include differentiating revenue (price times quantity), kinetic energy (mass times velocity squared), and wave functions in quantum mechanics. The product rule also underlies integration by parts, which reverses the product rule: integral of f dg = fg - integral of g df.

Formula Reference

Product Rule

d/dx[f*g] = f'*g + f*g'

Variables: f(x) = ax^m, g(x) = bx^n, f'(x) = amx^(m-1), g'(x) = bnx^(n-1)

Simplified Result for Power Functions

d/dx[ax^m * bx^n] = ab(m+n)x^(m+n-1)

Variables: a,b = coefficients; m,n = exponents

Worked Examples

Example 1: Derivative of 2x^3 * 5x^2

Find d/dx [2x^3 * 5x^2] using the product rule.

Step 1:Let f(x) = 2x^3, g(x) = 5x^2
Step 2:f'(x) = 6x^2, g'(x) = 10x
Step 3:Product rule: f'g + fg' = 6x^2 * 5x^2 + 2x^3 * 10x = 30x^4 + 20x^4 = 50x^4
Step 4:Or using the formula: ab(m+n)x^(m+n-1) = 2*5*(3+2)x^(4) = 50x^4

The derivative is 50x^4 (coefficient = 50, exponent = 4).

Example 2: Derivative of 3x * 4x^2

Find d/dx [3x * 4x^2].

Step 1:f = 3x (a=3, m=1), g = 4x^2 (b=4, n=2)
Step 2:Combined coefficient: ab(m+n) = 3*4*(1+2) = 36
Step 3:Combined exponent: m+n-1 = 1+2-1 = 2
Step 4:Result: 36x^2

The derivative is 36x^2.

Common Mistakes & Tips

  • !Thinking the derivative of a product is the product of the derivatives: d/dx[fg] is NOT f'*g'. You must use f'g + fg'.
  • !Forgetting one of the two terms in the product rule. The result has two addends: f'g AND fg'.
  • !Mixing up the product rule with the chain rule. The product rule applies to f(x)*g(x); the chain rule applies to f(g(x)).

Related Concepts

Quotient Rule

Differentiates a quotient f/g using [f'g - fg']/g^2. It is derived from the product rule applied to f * g^(-1).

Chain Rule

Differentiates composite functions f(g(x)). Often used together with the product rule on complex expressions.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Chain Rule Calculator

Derivative Calculator

Quotient Rule Calculator

Frequently Asked Questions

Why can't I just multiply the derivatives together?

Because differentiation does not distribute over multiplication. The product of derivatives f'(x)*g'(x) is generally not equal to the derivative of the product. The product rule correctly accounts for how both factors change simultaneously.

How does the product rule extend to three or more functions?

For three functions: d/dx[fgh] = f'gh + fg'h + fgh'. Each function takes its turn being differentiated while the others are held constant. For n functions, there are n terms in the sum.

What is the connection between the product rule and integration by parts?

Integration by parts reverses the product rule. Starting from d/dx[fg] = f'g + fg', integrating both sides gives fg = integral(f'g) + integral(fg'). Rearranging: integral(fg') = fg - integral(f'g). This is the integration by parts formula.

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