Questions: y=ln(ln(4x)) dy/dx=

Solution Steps

To find the derivative of the function \( y = \ln(\ln(4x)) \), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the outer function is \( \ln(u) \) and the inner function is \( \ln(4x) \). We will first find the derivative of the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to \( x \).

We start with the function given in the problem: \[ y = \ln(\ln(4x)) \]

To find the derivative \(\frac{dy}{dx}\), we apply the chain rule. The derivative of \(y\) with respect to \(x\) can be expressed as: \[ \frac{dy}{dx} = \frac{d}{dx} \left( \ln(\ln(4x)) \right) \] Using the chain rule, we differentiate the outer function \(\ln(u)\) where \(u = \ln(4x)\): \[ \frac{dy}{dx} = \frac{1}{\ln(4x)} \cdot \frac{d}{dx}(\ln(4x)) \]

Next, we differentiate the inner function \(\ln(4x)\): \[ \frac{d}{dx}(\ln(4x)) = \frac{1}{4x} \cdot \frac{d}{dx}(4x) = \frac{1}{4x} \cdot 4 = \frac{1}{x} \]

Now we substitute back into our expression for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{1}{\ln(4x)} \cdot \frac{1}{x} = \frac{1}{x \ln(4x)} \]

Final Answer