Problem 8 - Calculus Early Transcendentals

Question

Evaluate the following derivatives. $$\frac{d}{d x}(\ln (\ln x))$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function \(y = \ln(\ln x)\). Answer: The derivative of the function is \(y' = \frac{1}{x\ln x}\).

Step 1: Identify the composite function

The given function is \(\ln(\ln x)\). Notice that it is a composition of two functions: the outer function is \(\ln(u)\) and the inner function is \(u = \ln x\).

Step 2: Apply the chain rule

The chain rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function, and the derivative of the inner function. So, let's first evaluate the derivative of the outer function. Outer function is \(f(u) = \ln u\). Using the derivative of the natural logarithm: $$\frac{df}{du} = \frac{1}{u}$$ The inner function is \(u(x) = \ln x\). Again, using the derivative of the natural logarithm: $$\frac{du}{dx} = \frac{1}{x}$$ Now, applying the chain rule, we obtain: $$\frac{d}{dx} (\ln(\ln x)) = \frac{df}{du}\cdot\frac{du}{dx}$$

Step 3: Compute the derivative

Substitute the expressions for \(\frac{df}{du}\) and \(\frac{du}{dx}\) that we computed in step 2: $$\frac{d}{dx} (\ln(\ln x)) = \left(\frac{1}{u}\right)\left(\frac{1}{x}\right)$$ Now we need to substitute back \(u = \ln x\): $$\frac{d}{dx} (\ln(\ln x)) = \frac{1}{\ln x} \cdot \frac{1}{x}$$

Step 4: Simplify the result

We can write the final answer as a single fraction: $$\frac{d}{dx} (\ln(\ln x)) = \frac{1}{x\ln x}$$ So, the derivative of the given function is: $$\frac{d}{d x}(\ln (\ln x)) = \frac{1}{x\ln x}$$