Question

question number 11. (4.00 points)
we know $f(x)$ is a differentiable function. we are given its first derivative as:
$$f(x) = \frac{10\cos(10x)}{1 + \sin^2(10x)}$$
which of the following might be the formula for $f(x)$?
$\circ f(x) = \arcsin(\cos(10x))$

$\circ f(x) = \arctan(\cos 10x)$

$\circ f(x) = \arctan(\sin(10x))$

$\circ f(x) = \sin(10x)\arctan(x)$

$\circ f(x) = \arcsec(\sin(10x))$

$\circ$ none of the above.

Explanation:

To determine which function \( f(x) \) has the derivative \( f'(x) = \frac{10 \cos(10x)}{1 + \sin^2(10x)} \), we use the chain rule to differentiate each option.

Step 1: Recall the derivative of \( \arctan(u) \)

The derivative of \( \arctan(u) \) with respect to \( x \) is \( \frac{u'}{1 + u^2} \), where \( u \) is a function of \( x \).

Step 2: Differentiate \( f(x) = \arctan(\sin(10x)) \)

Let \( u = \sin(10x) \). Then, by the chain rule:

• First, find \( u' \): The derivative of \( \sin(10x) \) with respect to \( x \) is \( 10 \cos(10x) \) (using the chain rule again: derivative of \( \sin(v) \) is \( \cos(v) \cdot v' \), where \( v = 10x \), so \( v' = 10 \)).
• Now, apply the derivative formula for \( \arctan(u) \):

\[
f'(x) = \frac{u'}{1 + u^2} = \frac{10 \cos(10x)}{1 + (\sin(10x))^2} = \frac{10 \cos(10x)}{1 + \sin^2(10x)}
\]
This matches the given derivative \( f'(x) \).

We can quickly check the other options to confirm:

• For \( f(x) = \arcsin(\cos(10x)) \), the derivative would involve the derivative of \( \arcsin(u) \), which is \( \frac{u'}{\sqrt{1 - u^2}} \), and it will not match the given form.
• For \( f(x) = \arctan(\cos(10x)) \), the derivative would have \( -\sin(10x) \) in the numerator (since derivative of \( \cos(10x) \) is \( -10 \sin(10x) \)), which does not match.
• For \( f(x) = \sin(10x) \arctan(x) \), we would need to use the product rule, and the derivative will be more complex and not match the given form.
• For \( f(x) = \text{arcsec}(\sin(10x)) \), the derivative of \( \text{arcsec}(u) \) is \( \frac{u'}{|u| \sqrt{u^2 - 1}} \), which will not match the given form.

Answer:

\( \boldsymbol{f(x) = \arctan(\sin(10x))} \) (the third option)