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

Answer

To find \( \frac{dy}{dx} \) for the function \( y = \cos^4 x \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

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

### Step 2: Apply the chain rule
According to the chain rule, the derivative of \( y \) with respect to \( x \) can be expressed as:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( y \) with respect to \( u \)
First, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 4u^3 = 4(\cos x)^3 \]

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

### Step 5: Combine the derivatives
Now we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4(\cos x)^3 \cdot (-\sin x) \]

### Step 6: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = -4 \cos^3 x \sin x \]

### Final Result
The derivative of \( y = \cos^4 x \) is:
\[ \frac{dy}{dx} = -4 \cos^3 x \sin x \]

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