Find `dy/dx if y= sin^4x`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \sin^4 x\), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \sin^4 x \]
This can be viewed as a composition of functions where \(u = \sin x\) and \(y = u^4\).

### Step 2: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the outer function
First, we differentiate \(y = u^4\):
\[ \frac{dy}{du} = 4u^3 \]

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

### Step 5: Combine the derivatives
Now, substitute \(u = \sin x\) back into the derivative:
\[ \frac{dy}{dx} = 4(\sin x)^3 \cdot \cos x \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 4 \sin^3 x \cos x \]

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