Differentiate the following w.r.t.x. `sin^(3)x`

Answer

To differentiate the function \( y = \sin^3 x \) with respect to \( x \), we will use the chain rule and the basic differentiation formulas. Here’s a step-by-step solution:

### Step 1: Identify the function
Let \( y = \sin^3 x \).

### Step 2: Apply the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if \( y = u^n \) where \( u = \sin x \) and \( n = 3 \), then:

\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( y \) with respect to \( u \)
First, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 3u^2 = 3(\sin x)^2 \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Next, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \cos x \]

### Step 5: Combine the results
Now, we can combine the results from Steps 3 and 4:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3(\sin x)^2 \cdot \cos x \]

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

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