Problem 11 - Calculus Early Transcendentals

Question

Evaluate the following derivatives. $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right)$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function \((\ln(2x))^{-5}\) with respect to \(x\). Answer: The derivative of the function \((\ln(2x))^{-5}\) with respect to \(x\) is \(\frac{-5}{(2x)(\ln(2x))^6}\).

Step 1: Identify the Inner and Outer Functions

In this case, the inner function is \(\ln(2x)\) and the outer function is given by \(u^{-5}\), where \(u = \ln(2x)\). So we need to find the derivative of the outer function with respect to the inner function and also the derivative of the inner function with respect to \(x\).

Step 2: Differentiate the Inner Function

We have to find the derivative of \(\ln(2x)\) with respect to \(x\). The derivative of the natural logarithm function is given by the formula: \(\frac{d}{d x}\ln(x) = \frac{1}{x}\). So, in our case, the derivative of the inner function is: $$\frac{d}{d x}(\ln(2x)) = \frac{1}{2x}$$

Step 3: Differentiate the Outer Function

Now, we need to find the derivative of the outer function \(u^{-5}\) with respect to \(u\). To do this, we can use the power rule, which states that \(\frac{d}{d u}(u^n) = n \cdot u^{n-1}\). In our case, this gives us: $$\frac{d}{d u}(u^{-5}) = -5u^{-6}$$

Step 4: Apply the Chain Rule for Derivative

Now we can apply the chain rule, which states that for the derivative of a function \(h(x) = f\big(g(x)\big)\), we have \(\frac{dh}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}\). In our case, this gives us: $$\frac{d}{d x}((\ln(2x))^{-5}) = \left(-5(\ln(2x))^{-6}\right) \cdot \left(\frac{1}{2x}\right)$$

Step 5: Simplify the Expression

Finally, we can simplify the expression for the derivative. $$\frac{d}{d x}((\ln(2x))^{-5}) = \frac{-5}{(2x)(\ln(2x))^6}$$ So, the derivative of the given function is: $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right) = \frac{-5}{(2x)(\ln(2x))^6}$$