If [tex][tex]$f(x)=2\left(\sin^3 x\right)$[/tex][/tex], find [tex][tex]$f^{\prime}\left(\frac{\pi}{4}\right)$[/tex][/tex].

A. 2
B. 6
C. [tex][tex]$\frac{3}{\sqrt{2}}$[/tex][/tex]
D. 3



Answer :

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.