Derivative of sec^3 x: Answer, Proof, and Mistakes

Chain rule with the power on the outside

Read $\sec^{3}x$ as $\left(\sec x\right)^{3}$. The outer function is the cube, the inner function is $\sec x$, and the inner derivative is $\sec x\tan x$.

Now count secants. The power rule hands back $\sec^{2}x$, one power lower than you started with, and the inner derivative contributes one more. Two plus one is three, so the exponent returns to where it began.

Domain, shape and the integral that comes free

$\sec x$ is undefined wherever $\cos x = 0$, so both $f$ and $f'$ live on intervals such as $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. The sample inputs $x = 0.3, 0.7, 1.1$ all sit inside that interval.

On that interval $\sec^{3}x$ is positive, so the sign of $f'$ is the sign of $\tan x$. The function falls on $\left(-\frac{\pi}{2},0\right)$ and rises on $\left(0,\frac{\pi}{2}\right)$, with a minimum at $x=0$ of value $\sec^{3}0 = 1$.

Read the result backwards and it is an antiderivative in disguise: since $\frac{d}{dx}\left(\frac{1}{3}\sec^{3}x\right) = \sec^{3}x\tan x$, you get $\int \sec^{3}x\tan x\,dx = \frac{1}{3}\sec^{3}x+C$ without any work, which is the $u=\sec x$ substitution.

The mistakes students make

The first two wrong answers come from mishandling the inner derivative of sec x, and the third comes from misreading the notation.

• Answering $3\sec^{2}x\tan x$. That is the power rule multiplied by $\tan x$ alone. The inner derivative is $\sec x\tan x$, and the extra $\sec x$ pushes the power back up to $3$.
• Answering $3\sec^{2}x$, treating $\sec x$ as if it were the variable and stopping after the power rule. The chain rule is not optional once the inner function is anything other than $x$.
• Reading the function as $\sec\left(x^{3}\right)$ and answering $3x^{2}\sec\left(x^{3}\right)\tan\left(x^{3}\right)$. The notation $\sec^{3}x$ cubes the output, not the input.