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Chapter 6: Problem 11

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

Short Answer

Expert verified
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 by step solution

01

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 \(x\).
02

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: $$\frac{d}{d x}(\ln(2x)) = \frac{1}{2x}$$
03

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: $$\frac{d}{d u}(u^{-5}) = -5u^{-6}$$
04

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: $$\frac{d}{d x}((\ln(2x))^{-5}) = \left(-5(\ln(2x))^{-6}\right) \cdot \left(\frac{1}{2x}\right)$$
05

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: $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right) = \frac{-5}{(2x)(\ln(2x))^6}$$

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