Derivative of (x^2+1)^3: Answer, Proof, Mistakes

How to differentiate (x^2+1)^3

The outer function is the cube and the inner is $x^2+1$. Use $\frac{d}{dx}u^3 = 3u^2\,u'$ with $u = x^2+1$ and $u' = 2x$.

The coefficient $6$ is the product of the outer exponent $3$ and the inner derivative's coefficient $2$. Leave the answer factored; expanding the cube first is legal but slower and error-prone.

Where the derivative of (x^2+1)^3 shows up on the AP exam

This is a textbook chain rule item, Topic 3.1 on AB and BC: a polynomial raised to a power. The factored form $6x(x^2+1)^2$ is also convenient because you can read its sign directly. Since $(x^2+1)^2 > 0$ always, the derivative has the sign of $x$, so the function decreases for $x < 0$ and increases for $x > 0$, with a minimum at $x = 0$.

Keeping the derivative factored pays off in curve-sketching and optimization, where you set it to $0$ and $x = 0$ is the only critical point.

Common mistakes with the derivative of (x^2+1)^3

• Answering $3(x^2+1)^2$ and forgetting the inner derivative $2x$. That is the most common chain rule slip.
• Answering $3(2x)^2 = 12x^2$, differentiating the inside before applying the outer power. The inside stays intact under the cube.
• Writing $6x(x^2+1)^3$, keeping the original exponent $3$ instead of reducing it to $2$.