To find [tex]\( f'(x) \)[/tex] for the function [tex]\( f(x) = 2 (\sin^3 x) \)[/tex] and then evaluate [tex]\( f'( \frac{\pi}{4} ) \)[/tex], follow these steps:
1. Define the function and find its derivative:
The given function is [tex]\( f(x) = 2 (\sin^3 x) \)[/tex].
2. Use the chain rule to find [tex]\( f'(x) \)[/tex]:
Recall that if [tex]\( g(x) = \sin^3 x \)[/tex], then [tex]\( f(x) = 2 g(x) \)[/tex]. The chain rule tells us that:
[tex]\[
f'(x) = 2 \cdot \frac{d}{dx} (\sin^3 x)
\][/tex]
Now, let’s find the derivative of [tex]\( \sin^3 x \)[/tex]:
[tex]\[
\frac{d}{dx} (\sin^3 x) = 3 \sin^2 x \cdot \cos x
\][/tex]
Therefore,
[tex]\[
f'(x) = 2 \cdot 3 \sin^2 x \cdot \cos x = 6 \sin^2 x \cdot \cos x
\][/tex]
3. Evaluate the derivative at [tex]\( x = \frac{\pi}{4} \)[/tex]:
We need to substitute [tex]\( x = \frac{\pi}{4} \)[/tex] into the derivative:
[tex]\[
f'\left( \frac{\pi}{4} \right) = 6 \sin^2 \left( \frac{\pi}{4} \right) \cos \left( \frac{\pi}{4} \right)
\][/tex]
We know that [tex]\( \sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \)[/tex].
Substituting these values in:
[tex]\[
f'\left( \frac{\pi}{4} \right) = 6 \left( \frac{\sqrt{2}}{2} \right)^2 \cdot \frac{\sqrt{2}}{2}
\][/tex]
4. Simplify the expression:
First, calculate [tex]\( \left( \frac{\sqrt{2}}{2} \right)^2 \)[/tex]:
[tex]\[
\left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2}
\][/tex]
Now, substitute this back into the expression:
[tex]\[
f'\left( \frac{\pi}{4} \right) = 6 \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3 \sqrt{2}}{2}
\][/tex]
5. Convert the value into a decimal form for verification:
[tex]\[
\frac{3 \sqrt{2}}{2} \approx 2.12132034355964
\][/tex]
Therefore, the result of [tex]\( f'\left( \frac{\pi}{4} \right) \)[/tex] is [tex]\( \boxed{2.12132034355964} \)[/tex], which complies with the provided numerical result.
The provided choices are:
1. 2
2. 6
3. [tex]\( \frac{3}{\sqrt{2}} \)[/tex]
4. 3
The correct answer can be deemed as closely related to the integer answer based on the given approximations, indicating none of the given options exactly match the result derived.
