Find dy/dx if y = (ln(x))^(cos(x)).

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Step-by-Step Solution

Step 1: Take logarithm on both sides

To differentiate a function of the form f(x)^{g(x)}$, we use logarithmic differentiation. We start by taking the natural logarithm on both sides of the equation.

Step 2: Apply logarithm property

Using the logarithm property \ln(a^b) = b \ln a$, we can simplify the right-hand side of the equation.

Step 3: Differentiate both sides with respect to x

Now, we differentiate both sides of the equation with respect to x$. On the left side, we use the chain rule. On the right side, we will use the product rule.

Step 4: Apply product rule and chain rule

Applying the product rule \frac{d}{dx}(uv) = u'v + uv' to the right side, where u = \cos x and v = \ln(\ln x)$. We also use the chain rule for differentiating \ln(\ln x)$.

Step 5: Isolate dy/dx

To find \frac{dy}{dx}$, we multiply both sides of the equation by y$.

Step 6: Substitute y back into the equation

Finally, we substitute the original expression for y$, which is (\ln x)^{\cos x}$, back into the equation to get the derivative in terms of x only.