Find `dy/dx if y= sin^2x`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \sin^2 x\), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
In the function \(y = \sin^2 x\), we can identify:
- Outer function: \(u^2\) where \(u = \sin x\)
- Inner function: \(u = \sin x\)

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \(u^2\) with respect to \(u\) is:
\[ \frac{dy}{du} = 2u \]

### Step 3: Differentiate the inner function
Now we differentiate the inner function \(u = \sin x\):
\[ \frac{du}{dx} = \cos x \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 2u \cdot \cos x \]

### Step 5: Substitute back the inner function
Now substitute \(u = \sin x\) back into the equation:
\[ \frac{dy}{dx} = 2 \sin x \cdot \cos x \]

### Step 6: Simplify using a trigonometric identity
The expression \(2 \sin x \cos x\) can be simplified using the double angle identity for sine:
\[ 2 \sin x \cos x = \sin(2x) \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \sin(2x) \]

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