Differentiate w.r.t x
`sin^4x`

Answer

To differentiate the function \( y = \sin^4 x \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
Let:
\[ y = \sin^4 x \]

### Step 2: Apply the chain rule
According to the chain rule, if we have a function \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, we can let:
\[ f(u) = u^4 \quad \text{where } u = \sin x \]
Thus, we have:
\[ y = f(u) = u^4 \]

### Step 3: Differentiate the outer function
Now, we differentiate \( f(u) \) with respect to \( u \):
\[ f'(u) = 4u^3 \]

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

### Step 5: Combine the derivatives using the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = f'(u) \cdot \frac{du}{dx} = 4u^3 \cdot \cos x \]
Substituting back \( u = \sin x \):
\[ \frac{dy}{dx} = 4(\sin x)^3 \cdot \cos x \]

### Step 6: Write the final answer
Thus, the derivative of \( y = \sin^4 x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 4 \sin^3 x \cos x \]