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.
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.