# Derivative of $\sec^{3}{\left(x \right)}$

The calculator will find the derivative of $\sec^{3}{\left(x \right)}$, with steps shown.

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Find $\frac{d}{dx} \left(\sec^{3}{\left(x \right)}\right)$.

### Solution

The function $\sec^{3}{\left(x \right)}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = u^{3}$ and $g{\left(x \right)} = \sec{\left(x \right)}$.

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$:

$${\color{red}\left(\frac{d}{dx} \left(\sec^{3}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)}$$

Apply the power rule $\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$ with $n = 3$:

$${\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right)$$

$$3 {\color{red}\left(u\right)}^{2} \frac{d}{dx} \left(\sec{\left(x \right)}\right) = 3 {\color{red}\left(\sec{\left(x \right)}\right)}^{2} \frac{d}{dx} \left(\sec{\left(x \right)}\right)$$

The derivative of the secant is $\frac{d}{dx} \left(\sec{\left(x \right)}\right) = \tan{\left(x \right)} \sec{\left(x \right)}$:

$$3 \sec^{2}{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)} = 3 \sec^{2}{\left(x \right)} {\color{red}\left(\tan{\left(x \right)} \sec{\left(x \right)}\right)}$$

Thus, $\frac{d}{dx} \left(\sec^{3}{\left(x \right)}\right) = 3 \tan{\left(x \right)} \sec^{3}{\left(x \right)}$.

$\frac{d}{dx} \left(\sec^{3}{\left(x \right)}\right) = 3 \tan{\left(x \right)} \sec^{3}{\left(x \right)}$A