Question

If $y = \sqrt{\cos(5x)}$, then $\frac{dy}{dx}$ is:

Answer 1

$\frac{-5\sin(5x)}{2\sqrt{\cos(5x)}}$

Answer 2

To find the derivative of $y = \sqrt{\cos(5x)}$ with respect to $x$, we use the chain rule.

The function can be written as $y = (\cos(5x))^{1/2}$.

We apply the chain rule in multiple steps:

1. Derivative of the outermost function (square root):
   Let $u = \cos(5x)$. Then $y = u^{1/2}$.
   The derivative of $y$ with respect to $u$ is:
   $\frac{dy}{du} = \frac{1}{2}u^{(1/2) - 1} = \frac{1}{2}u^{-1/2} = \frac{1}{2\sqrt{u}}$
   Substituting $u = \cos(5x)$ back, we get:
   $\frac{dy}{du} = \frac{1}{2\sqrt{\cos(5x)}}$

2. Derivative of the inner function (cosine):
   We need to find the derivative of $u = \cos(5x)$ with respect to $x$.
   Let $v = 5x$. Then $u = \cos(v)$.
   The derivative of $u$ with respect to $v$ is:
   $\frac{du}{dv} = -\sin(v)$
   Substituting $v = 5x$ back, we get:
   $\frac{du}{dv} = -\sin(5x)$

3. Derivative of the innermost function (linear term):
   We need to find the derivative of $v = 5x$ with respect to $x$.
   $\frac{dv}{dx} = 5$

4. Combine using the chain rule:
   According to the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$.
   Substituting the derivatives we found:
   $\frac{dy}{dx} = \left(\frac{1}{2\sqrt{\cos(5x)}}\right) \cdot (-\sin(5x)) \cdot (5)$

   Multiply these terms together:
   $\frac{dy}{dx} = \frac{-5\sin(5x)}{2\sqrt{\cos(5x)}}$

The final answer is $\frac{-5\sin(5x)}{2\sqrt{\cos(5x)}}$.

Explanation of the solution:

The derivative of $y = \sqrt{\cos(5x)}$ is found using the chain rule.

1. Differentiate the square root function: $\frac{d}{dx}(\sqrt{f(x)}) = \frac{1}{2\sqrt{f(x)}} \cdot f'(x)$.
   Here, $f(x) = \cos(5x)$. So, the first step gives $\frac{1}{2\sqrt{\cos(5x)}} \cdot \frac{d}{dx}(\cos(5x))$.
2. Differentiate $\cos(5x)$: This is another application of the chain rule.
   $\frac{d}{dx}(\cos(ax)) = -a\sin(ax)$.
   For $\cos(5x)$, $a=5$, so $\frac{d}{dx}(\cos(5x)) = -5\sin(5x)$.
3. Combine the results: Multiply the two parts obtained in steps 1 and 2.
   $\frac{dy}{dx} = \frac{1}{2\sqrt{\cos(5x)}} \cdot (-5\sin(5x)) = \frac{-5\sin(5x)}{2\sqrt{\cos(5x)}}$.