Questions: Differentiate V=sqrt(cos(8x))

Solution Steps

To differentiate the function \( V = \sqrt{\cos(8x)} \), we will use the chain rule. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is the square root function and the inner function is \( \cos(8x) \).

1. Identify the outer function and the inner function.
2. Differentiate the outer function with respect to the inner function.
3. Differentiate the inner function with respect to \( x \).
4. Multiply the results from steps 2 and 3.

We start with the function defined as: \[ V = \sqrt{\cos(8x)} \]

To find the derivative \( \frac{dV}{dx} \), we apply the chain rule. The derivative of the outer function \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \), where \( u = \cos(8x) \). Thus, we have: \[ \frac{dV}{dx} = \frac{1}{2\sqrt{\cos(8x)}} \cdot \frac{d}{dx}(\cos(8x)) \]

Next, we differentiate the inner function \( \cos(8x) \). The derivative of \( \cos(8x) \) is: \[ \frac{d}{dx}(\cos(8x)) = -8\sin(8x) \]

Substituting the derivative of the inner function back into our expression for \( \frac{dV}{dx} \), we get: \[ \frac{dV}{dx} = \frac{1}{2\sqrt{\cos(8x)}} \cdot (-8\sin(8x)) = -\frac{4\sin(8x)}{\sqrt{\cos(8x)}} \]

Final Answer