Derivative of `sinx^(3)` w.r.t. x is :

Options

A. `cosx^(3)`
B. `3x^(2)cosx^(3)`
C. `3x^(2)cosx`
D. None of these

Answer

To find the derivative of \( \sin(x^3) \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:

\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]

In our case, let:
- \( f(u) = \sin(u) \) where \( u = x^3 \)

### Step 1: Differentiate the outer function
First, we need to differentiate \( f(u) = \sin(u) \):
\[ f'(u) = \cos(u) \]

### Step 2: Differentiate the inner function
Next, we differentiate the inner function \( g(x) = x^3 \):
\[ g'(x) = 3x^2 \]

### Step 3: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{d}{dx} \sin(x^3) = f'(g(x)) \cdot g'(x) = \cos(x^3) \cdot 3x^2 \]

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

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