# What is the derivative of square root cos(x)?

The derivative of square root

\cos(x) is

\frac{-\sin(x)}{2\sqrt{\cos(x)}}.

## Solution of the derivative of square root cos(x)

Let

F(x) = f(g(x)) = \sqrt{\cos(x)}, where

f(u) = \sqrt{u} and

g(x) = \cos(x). To find the derivative of

\sqrt{\cos(x)}, we need to apply the chain rule:

\begin{align*}
F'(x) = f'(g(x))g'(x).
\end{align*}

We know from

here that

f'(u) = \frac{1}{2\sqrt{u}} and

here that

g'(x) = -\sin(x). Therefore, we get:

\begin{align*}
f'(g(x)) = \frac{1}{2\sqrt{g(x)}} = \frac{1}{2\sqrt{\cos(x)}} \quad \text{and} \quad g'(x) = -\sin(x).
\end{align*}

Finally, this gives us the desired result:

\begin{align*}
F'(x) &= f'(g(x))g'(x) \\
&= \frac{1}{2\sqrt{g(x)}} \cdot (-\sin(x)) \\
&= \frac{-\sin(x)}{2\sqrt{\cos(x)}}.
\end{align*}

## Conclusion

Therefore, the derivative of square root

\sqrt{\cos(x)} is

\frac{-\sin(x)}{2\sqrt{\cos(x)}}.