What if I have a product of three functions?

Apply the product rule sequentially or use the generalized product rule: $(uvw)' = u'vw + uv'w + uvw'$.

Does the order of multiplication matter?

No, multiplication is commutative, and addition in the product rule formula is also commutative. The expression $u'v + uv'$ is algebraically equivalent to $v'u + vu'$.

Can I use the product rule for a constant times a function?

Yes, but it is inefficient. Because the derivative of a constant is zero, one term in the product rule will vanish. The constant multiple rule is faster and yields the identical result.

How to differentiate a product of two functions

The product rule dictates that the derivative of a product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first. Apply this method when differentiating an expression composed of two multiplied variable terms, such as $x^2 \sin(x)$ .

The setup

Identify the expression as a product of two distinct functions. Assign one function to $u(x)$ and the other to $v(x)$ . Verify that both $u(x)$ and $v(x)$ are differentiable with respect to $x$ .

The steps

Define $u(x)$ and $v(x)$ from the original function $f(x) = u(x)v(x)$ .
Compute the derivative of the first function, $u'(x)$ .
Compute the derivative of the second function, $v'(x)$ .
Substitute these components into the product rule formula: $f'(x) = u(x)v'(x) + v(x)u'(x)$ .
Simplify the resulting algebraic expression by factoring or combining like terms.

Checking the result

If the functions are polynomials, expand the original expression algebraically before differentiating to verify that the standard power rule yields the same derivative. For more complex functions, verify the derivative at a specific point $x=a$ using a numerical derivative approximation.

Common errors

The most frequent error is assuming that the derivative of a product is the product of the derivatives, $f'(x) eq u'(x)v'(x)$ . Another common mistake is dropping negative signs when differentiating trigonometric or exponential components within $u(x)$ or $v(x)$ .

Worked example

Find the derivative of $f(x) = x^3 \cos(x)$ .

Let $u(x) = x^3$ and $v(x) = \cos(x)$ . Compute $u'(x) = 3x^2$ . Compute $v'(x) = -\sin(x)$ . Apply the product rule formula: $f'(x) = u(x)v'(x) + v(x)u'(x)$ . Substitute the terms: $f'(x) = (x^3)(-\sin(x)) + (\cos(x))(3x^2)$ . Simplify the expression: $f'(x) = -x^3 \sin(x) + 3x^2 \cos(x)$ .

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References: OpenStax Calculus Volume 1, Chapter 3 · James Stewart Calculus, Chapter 3