# How do you find the derivative of cos^2(x^3)?

Jan 26, 2016

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left({x}^{3}\right) = - 6 {x}^{2} \cos \left({x}^{3}\right) \sin \left({x}^{3}\right)$

#### Explanation:

The first issue here is the fact that the cosine function is squared, which maybe be misleading, since you want to deal with the cosine immediately.

Still, we can use the chain rule to say that $\frac{d}{\mathrm{dx}} {u}^{2} = 2 u \cdot u '$. Here, we have $u = \cos \left({x}^{3}\right)$.

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left({x}^{3}\right) = 2 \cos \left({x}^{3}\right) \cdot \frac{d}{\mathrm{dx}} \cos \left({x}^{3}\right)$

Now, when dealing with the cosine function's derivative, recall that $\frac{d}{\mathrm{dx}} \cos u = - \sin u \cdot u '$. We know that $u = {x}^{3}$.

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left({x}^{3}\right) = 2 \cos \left({x}^{3}\right) \left(- \sin \left({x}^{3}\right)\right) \frac{d}{\mathrm{dx}} {x}^{3}$

Since $\frac{d}{\mathrm{dx}} {x}^{3} = 3 {x}^{2}$ through the power rule, this leaves a final answer of

$\frac{d}{\mathrm{dx}} {\cos}^{2} \left({x}^{3}\right) = - 6 {x}^{2} \cos \left({x}^{3}\right) \sin \left({x}^{3}\right)$