Derivative of square root cos(x)

The Mathematician — Thu, 18 May 2023 13:00:24 +0000

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

The derivative of square root

\cos(x)

\frac{-\sin(x)}{2\sqrt{\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*}

Therefore, the derivative of square root

\sqrt{\cos(x)}

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