If `f(x)=sin^(-1)(3x-4x^(3)).` Then answer the following
The value of `f'((1)/(sqrt2))` , is

Options

A. -3
B. 3
C. `-3sqrt2`
D. `3sqrt2`

Answer

To solve the problem, we need to find the derivative of the function \( f(x) = \sin^{-1}(3x - 4x^3) \) and evaluate it at \( x = \frac{1}{\sqrt{2}} \).

### Step-by-Step Solution:

1. **Differentiate the Function**:
We start with the function:
\[ f(x) = \sin^{-1}(3x - 4x^3) \]
To find the derivative \( f'(x) \), we use the chain rule:
\[ f'(x) = \frac{1}{\sqrt{1 - (3x - 4x^3)^2}} \cdot \frac{d}{dx}(3x - 4x^3) \]
Now, we differentiate \( 3x - 4x^3 \):
\[ \frac{d}{dx}(3x - 4x^3) = 3 - 12x^2 \]
Therefore, we have:
\[ f'(x) = \frac{3 - 12x^2}{\sqrt{1 - (3x - 4x^3)^2}} \]

2. **Evaluate at \( x = \frac{1}{\sqrt{2}} \)**:
Now we substitute \( x = \frac{1}{\sqrt{2}} \) into the derivative:
\[ f'\left(\frac{1}{\sqrt{2}}\right) = \frac{3 - 12\left(\frac{1}{\sqrt{2}}\right)^2}{\sqrt{1 - (3\left(\frac{1}{\sqrt{2}}\right) - 4\left(\frac{1}{\sqrt{2}}\right)^3)^2}} \]
First, calculate \( \left(\frac{1}{\sqrt{2}}\right)^2 \):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Now substitute this into the expression:
\[ f'\left(\frac{1}{\sqrt{2}}\right) = \frac{3 - 12 \cdot \frac{1}{2}}{\sqrt{1 - (3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \frac{1}{2\sqrt{2}})^2}} \]
Simplifying the numerator:
\[ 3 - 12 \cdot \frac{1}{2} = 3 - 6 = -3 \]

3. **Calculate the Denominator**:
Next, we calculate the expression inside the square root in the denominator:
\[ 3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \frac{1}{2\sqrt{2}} = \frac{3}{\sqrt{2}} - \frac{2}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
Now square this result:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Thus, the denominator becomes:
\[ \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

4. **Final Calculation**:
Now we can substitute back into our expression for \( f'\left(\frac{1}{\sqrt{2}}\right) \):
\[ f'\left(\frac{1}{\sqrt{2}}\right) = \frac{-3}{\frac{1}{\sqrt{2}}} = -3 \cdot \sqrt{2} = -3\sqrt{2} \]

### Final Answer:
\[ f'\left(\frac{1}{\sqrt{2}}\right) = -3\sqrt{2} \]