# 2.2.206 AP Calculus Practice question derivative of a composite sine

• MHB
• karush
In summary, the conversation discusses finding the derivative of the function $f(x)=\sin{(\ln{(2x)})}$. After inputting the function into Wolfram|Alpha, the result returned is (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$, but the questioner is unsure why the $\ln(2x)$ was not changed. The expert explains that the chain rule is used to find the derivative, and the argument of the derivative is still $\ln{(2x)}$, not the derivative itself.
karush
Gold Member
MHB
206 (day of year number)

If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$
(A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$

(B) $\dfrac{\cos{(\ln{(2x)}}}{x}$

(C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$

(D) $\cos{\left(\dfrac{1}{2x}\right)}$

Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$
but I didn't understand why the $\ln(2x)$ was not changed?

these are being also posted in MeWe in the MathQuiz group

$$\displaystyle g(x)=\log(2x)=\log(2)+\log(x),\quad g'(x)=\frac1x$$

or, by the chain rule,

$$\displaystyle g'(x)=\frac{2}{2x}=\frac1x$$

By the chain rule, $$f(g(x))'= f'(g(x))g'(x)$$. Notice that the argument of f' is still g(x), not the derivative.

## 1. What is a composite function?

A composite function is a function that is made up of two or more simpler functions. The output of one function is used as the input for the other function.

## 2. How do you find the derivative of a composite function?

To find the derivative of a composite function, you can use the chain rule. This involves taking the derivative of the outer function and multiplying it by the derivative of the inner function.

## 3. What is the derivative of a sine function?

The derivative of a sine function is a cosine function. This can be written as f'(x) = cos(x).

## 4. How do you apply the chain rule to find the derivative of a composite sine function?

To apply the chain rule, you first identify the outer function and the inner function. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function.

## 5. Can you provide an example of finding the derivative of a composite sine function?

Yes, for example, if we have the function f(x) = sin(3x), we can use the chain rule to find the derivative. First, we identify the outer function as sin(x) and the inner function as 3x. Then, we take the derivative of the outer function, which is cos(x), and multiply it by the derivative of the inner function, which is 3. Therefore, the derivative of f(x) = sin(3x) is f'(x) = 3cos(3x).

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