Compute the derivative of `f(x)=sin^2x`.

Options

A. `cos^(2) x`
B. 2 sin x
C. sin 2 X
D. cos 2 x

Answer

To compute the derivative of the function \( f(x) = \sin^2 x \), we can use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
We can express \( f(x) \) as:
\[ f(x) = (\sin x)^2 \]
Here, the outer function is \( u^2 \) where \( u = \sin x \).

### Step 2: Differentiate the outer function
The derivative of \( u^2 \) with respect to \( u \) is:
\[ \frac{d}{du}(u^2) = 2u \]

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

### Step 4: Apply the chain rule
Using the chain rule, we can find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{du}(u^2) \cdot \frac{du}{dx} = 2u \cdot \cos x \]
Substituting back \( u = \sin x \):
\[ f'(x) = 2(\sin x)(\cos x) \]

### Step 5: Simplify the expression
We can also express this result using the double angle identity:
\[ f'(x) = \sin(2x) \]

### Final Answer
Thus, the derivative of \( f(x) = \sin^2 x \) is:
\[ f'(x) = \sin(2x) \]

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