`(d)/(dx)(sinx-(1)/(3)sin^(3)x)=`

Options

A. `sin^(2)x`
B. `cos^(2)x`
C. `-sin^(3)x`
D. `(3cosx+cos3x)//4`

Answer

To find the derivative of the function \( f(x) = \sin x - \frac{1}{3} \sin^3 x \), we will apply the rules of differentiation step by step.

### Step 1: Differentiate \( \sin x \)

The derivative of \( \sin x \) is \( \cos x \).

\[ \frac{d}{dx}(\sin x) = \cos x \]

### Step 2: Differentiate \( -\frac{1}{3} \sin^3 x \)

To differentiate \( -\frac{1}{3} \sin^3 x \), we will use the chain rule. The derivative of \( u^n \) is \( n u^{n-1} \cdot \frac{du}{dx} \), where \( u = \sin x \) and \( n = 3 \).

1. Differentiate the outer function \( -\frac{1}{3} u^3 \):
\[ \frac{d}{dx}\left(-\frac{1}{3} \sin^3 x\right) = -\frac{1}{3} \cdot 3 \sin^2 x \cdot \frac{d}{dx}(\sin x) \]
This simplifies to:
\[ -\sin^2 x \cdot \cos x \]

### Step 3: Combine the results

Now we combine the derivatives from Step 1 and Step 2:

\[ \frac{d}{dx}\left(\sin x - \frac{1}{3} \sin^3 x\right) = \cos x - \sin^2 x \cdot \cos x \]

### Step 4: Factor out \( \cos x \)

We can factor out \( \cos x \) from the expression:

\[ \cos x - \sin^2 x \cdot \cos x = \cos x (1 - \sin^2 x) \]

### Step 5: Simplify using the Pythagorean identity

Using the Pythagorean identity \( 1 - \sin^2 x = \cos^2 x \), we can simplify further:

\[ \cos x (1 - \sin^2 x) = \cos x \cdot \cos^2 x \]

### Final Answer

Thus, the derivative is:

\[ \frac{d}{dx}\left(\sin x - \frac{1}{3} \sin^3 x\right) = \cos^3 x \]

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