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.
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
How to Use This Calculator
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.
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.
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.
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.
Formula Reference
Product Rule Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
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.
About This Calculator
The Product Rule Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. 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. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
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
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.
The derivative is 50x^4 (coefficient = 50, exponent = 4).
Example 2: Derivative of 3x * 4x^2
Find d/dx [3x * 4x^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
Used in These Calculators
Calculators that build on or apply the concepts from this page:
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.