If `y=cosx^3`, then find `(dy)/(dx)`.

Options

A. `3x^2sinx^3`
B. `-3x^2cosx^3`
C. `-3x^2sinx^2`
D. `-3x^2sinx^3`

Explanation

To find the derivative of the function \( y = \cos(x^3) \) with respect to \( x \), we can use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
The function \( y = \cos(x^3) \) can be viewed as a composition of two functions:
- Outer function: \( u = \cos(t) \) where \( t = x^3 \)
- Inner function: \( t = x^3 \)

### Step 2: Differentiate the outer function
The derivative of the outer function \( u = \cos(t) \) with respect to \( t \) is:
\[ \frac{du}{dt} = -\sin(t) \]

### Step 3: Differentiate the inner function
The derivative of the inner function \( t = x^3 \) with respect to \( x \) is:
\[ \frac{dt}{dx} = 3x^2 \]

### Step 4: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{du}{dt} \cdot \frac{dt}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(t) \cdot 3x^2 \]

### Step 5: Substitute back the inner function
Now, we substitute \( t = x^3 \) back into the equation:
\[ \frac{dy}{dx} = -\sin(x^3) \cdot 3x^2 \]

### Final Answer
Thus, the derivative of \( y = \cos(x^3) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -3x^2 \sin(x^3) \]

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