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 then evaluate it at \( x = \frac{1}{\sqrt{2}} \).

### Step-by-step Solution:

**Step 1: Differentiate \( f(x) \)**

We start with the function:
\[ f(x) = \sin^{-1}(3x - 4x^3) \]

Using the chain rule, the derivative of \( \sin^{-1}(u) \) is given by:
\[ \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]

Here, \( u = 3x - 4x^3 \). We first need to find \( \frac{du}{dx} \).

**Step 2: Find \( \frac{du}{dx} \)**

Differentiate \( u \):
\[ u = 3x - 4x^3 \]
\[ \frac{du}{dx} = 3 - 12x^2 \]

**Step 3: Substitute into the derivative formula**

Now substituting \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ f'(x) = \frac{1}{\sqrt{1 - (3x - 4x^3)^2}} \cdot (3 - 12x^2) \]

**Step 4: Evaluate \( f' \left( \frac{1}{\sqrt{2}} \right) \)**

Now we need to evaluate \( f' \left( \frac{1}{\sqrt{2}} \right) \):
\[ f' \left( \frac{1}{\sqrt{2}} \right) = \frac{1}{\sqrt{1 - (3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^3)^2}} \cdot \left(3 - 12 \cdot \left(\frac{1}{\sqrt{2}}\right)^2\right) \]

**Step 5: Simplify the expression inside the square root**

Calculate \( 3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^3 \):
\[ 3 \cdot \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]
\[ 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^3 = 4 \cdot \frac{1}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]
Thus,
\[ 3 \cdot \frac{1}{\sqrt{2}} - 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^3 = \frac{3}{\sqrt{2}} - \sqrt{2} = \frac{3 - 2}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

Now, square this result:
\[ \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2} \]

**Step 6: Substitute back into the derivative**

Now substitute this back into the derivative:
\[ f' \left( \frac{1}{\sqrt{2}} \right) = \frac{1}{\sqrt{1 - \frac{1}{2}}} \cdot \left(3 - 12 \cdot \frac{1}{2}\right) \]
\[ = \frac{1}{\sqrt{\frac{1}{2}}} \cdot \left(3 - 6\right) = \frac{1}{\frac{1}{\sqrt{2}}} \cdot (-3) = -3\sqrt{2} \]

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