Derivative of $\sec^2(x)$
By Michael Henderson •
I know that the derivative of $\sec(x)$ is $\sec(x)\tan(x)$, so what is the derivative of $sec^2(x)$?
I want to know what derivative rules you use to derive it.
$\endgroup$ 23 Answers
$\begingroup$Remember $\sec^2 x=1/\cos^2 x$.
You can use the chain rule to take the derivative: $$ \frac{d}{dx}(\cos x)^{-2}=-2(\cos x)^{-3}(-\sin x)=2\tan x\sec^2 x. $$
$\endgroup$ 1 $\begingroup$$$\frac{d}{dx}\sec^2(x)$$ Apply chain rule, setting $u= \sec(x)$ $$=\frac{du^2}{du}\frac{du}{dx}$$ $$=2u\frac{d}{dx}\sec(x)$$ $$=2\sec(x)\sec(x)\tan(x)$$ $$=2\sec^2(x)\tan(x)$$
$\endgroup$ $\begingroup$Unfortunately in standard notation $\sec^2 x$ means $\Big(\sec x\Big)^2$ rather than $\sec(\sec x)$. Therefore $$ \frac d {dx} \sec^2 x = \frac d {dx} \Big((\sec x)^2\Big) = 2\sec x \cdot \frac d {dx} \sec x= \cdots. $$
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