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Chapter 6: Problem 9

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

Short Answer

Expert verified
Answer: The derivative of the function $$\sin(\ln(x))$$ with respect to $$x$$ is $$\frac{\cos(\ln(x))}{x}$$.

Step by step solution

01

Identify the Outer and Inner Functions

The given function is $$\sin(\ln(x))$$. The outer function is $$\sin(u)$$, where $$u = \ln(x)$$. And the inner function is $$\ln(x)$$.
02

Derivative of the Outer Function

Now, differentiate the outer function with respect to $$u$$: $$\frac{d}{d u}(\sin(u)) = \cos(u)$$. We are treating $$u$$ as an independent variable in this part.
03

Derivative of the Inner Function

Next, differentiate the inner function with respect to $$x$$: $$\frac{d}{d x}(\ln(x)) = \frac{1}{x}$$.
04

Apply the Chain Rule

Using the chain rule, multiply the derivatives calculated in Steps 2 and 3: $$\frac{d}{d x}(\sin(\ln(x))) = \frac{d}{d u}(\sin(u)) \cdot \frac{d u}{d x} = \cos(u) \cdot \frac{1}{x}$$.
05

Substitute $$u$$ back and Simplify

Finally, substitute $$u = \ln(x)$$ back into the expression: $$\frac{d}{d x}(\sin(\ln(x))) = \cos(\ln(x)) \cdot \frac{1}{x}$$. The derivative of $$\sin(\ln(x))$$ with respect to $$x$$ is $$\frac{\cos(\ln(x))}{x}$$.

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