Product Rule Derivative Calculator

Differentiate products of two functions using the product rule: (u·v)' = u'·v + u·v'. Step‑by‑step breakdown and graphical visualization.

Product Rule: d/dx [ u(x) · v(x) ] = u'(x) · v(x) + u(x) · v'(x)

Enter the two functions below.

x² · sin(x) eˣ · cos(x) ln(x) · x³ sin(x)·cos(x) x·eˣ
Computing...

Understanding the Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states:

(f·g)'(x) = f'(x)·g(x) + f(x)·g'(x)

Derivation from First Principles

Starting from the limit definition of the derivative:

(f·g)'(x) = limh→0 [f(x+h)g(x+h) - f(x)g(x)] / h

Add and subtract f(x+h)g(x) in the numerator:

= limh→0 [f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)] / h

= limh→0 f(x+h)·[g(x+h)-g(x)]/h + limh→0 [f(x+h)-f(x)]/h · g(x)

Assuming f and g are differentiable, we obtain f(x)g'(x) + f'(x)g(x).

Geometric Intuition: Area of a Rectangle

Think of u·v as the area of a rectangle with sides u and v. A small change dx changes u by du and v by dv. The new area is (u+du)(v+dv) = uv + u·dv + v·du + du·dv. The change in area (the derivative) is u·dv + v·du, ignoring the second‑order term du·dv. This is exactly the product rule.

Common Examples

u(x) v(x) Product derivative (simplified)
x² sin x 2x sin x + x² cos x
eˣ cos x eˣ(cos x - sin x)
ln x x³ 3x² ln x + x²
sin x cos x cos²x - sin²x = cos 2x
x eˣ eˣ(x + 1)

Extended Product Rule (Three Factors)

For three functions f, g, h, the product rule becomes:

(fgh)' = f'gh + fg'h + fgh'

This pattern continues for any number of factors: the derivative is the sum of the derivative of each factor times the product of the remaining factors.

Relationship with the Quotient Rule

The quotient rule can be derived by writing f/g = f·(1/g) and applying the product rule together with the chain rule:

(f/g)' = f'·(1/g) + f·(-1/g²)·g' = (f'g - fg')/g².

Common Mistakes

  • Forgetting to add both terms – the product rule requires both u'v and uv'.
  • Misidentifying which part is u and which is v – the rule is symmetric, but correct differentiation of each part is essential.
  • Not simplifying the final expression – often the result can be factored or rewritten in a more compact form.

Applications

  • Physics: derivative of momentum (mass·velocity) when mass varies (rocket motion).
  • Economics: marginal revenue as derivative of price·quantity.
  • Biology: population models with interacting species.
  • Engineering: signal modulation (product of two signals).

Calculator features:

  • Symbolic differentiation using nerdamer.js for exact results.
  • Step‑by‑step display of u, v, their derivatives, and the product rule application.
  • Optional evaluation at a specific x‑value.
  • Graph of the product function and its derivative.
  • Supports mathematical constants like π and e in expressions.

Frequently Asked Questions

Yes, there is an extended product rule for three or more functions: (fgh)' = f'gh + fg'h + fgh'. You can apply it repeatedly.

The constant multiple rule applies: (c·f)' = c·f'. You don't need the product rule in that case. However, if you treat the constant as a function, the product rule still works (since derivative of constant is zero).

The quotient rule can be derived by writing f/g = f·(1/g) and applying the product rule together with the chain rule. It gives (f/g)' = (f'g - fg')/g².

Yes, √x can be entered as sqrt(x). For absolute value, use abs(x). Nerdamer supports these functions, and the derivative will be computed symbolically where possible.