If `y = cos^(-1) (4x^(3) -3x) " then " (dy)/(dx)=` ?

Options

A. `(3)/(sqrt(1 -x^(2)))`
B. `(-3)/(sqrt(1 -x^(2)))`
C. `(4)/(sqrt(1 -x^(2)))`
D. `(-4)/((3x^(2) -1))`

Explanation

To find the derivative of the function \( y = \cos^{-1}(4x^3 - 3x) \), we can follow these steps:

### Step 1: Identify the function
We have:
\[ y = \cos^{-1}(u) \quad \text{where} \quad u = 4x^3 - 3x \]

### Step 2: Differentiate using the chain rule
Using the chain rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( y \) with respect to \( u \)
The derivative of \( y = \cos^{-1}(u) \) is:
\[ \frac{dy}{du} = -\frac{1}{\sqrt{1 - u^2}} \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Now we differentiate \( u = 4x^3 - 3x \):
\[ \frac{du}{dx} = 12x^2 - 3 \]

### Step 5: Combine the derivatives
Now we can substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule formula:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - (4x^3 - 3x)^2}} \cdot (12x^2 - 3) \]

### Step 6: Final expression
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{12x^2 - 3}{\sqrt{1 - (4x^3 - 3x)^2}} \]

### Summary of the solution:
The final answer is:
\[ \frac{dy}{dx} = -\frac{12x^2 - 3}{\sqrt{1 - (4x^3 - 3x)^2}} \]

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