# What is the derivative of y=sec^3(x)?

##### 1 Answer
Aug 6, 2014

$\sec x$ is equal to $\frac{1}{\cos} x$.

Thus, ${\sec}^{3} x$ is an equivalent statement to $\frac{1}{\cos x} ^ 3$.

And, we know that this is equivalent to ${\left(\cos x\right)}^{- 3}$.

So, all we need to do is use the power rule, keeping in mind to use the chain rule on $\cos x$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 3 \cdot {\left(\cos x\right)}^{- 4} \cdot \frac{d}{\mathrm{dx}} \left[\cos x\right]$

$= - 3 \cdot {\left(\cos x\right)}^{- 4} \cdot \left(- \sin x\right)$

$= 3 \sin x {\left(\cos x\right)}^{- 4}$

$= \frac{3 \sin x}{{\cos}^{4} x}$