Find the derivatives of y = sin 4x

To find the derivative of \( y = \sin 4x \), you need to use the chain rule. The chain rule helps you take the derivative of a function inside another function.

First, recall that the derivative of \( \sin u \) with respect to \( u \) is \( \cos u \):

\[ \frac{d}{du} (\sin u) = \cos u \]

In this problem, \( u = 4x \). So when you differentiate \( \sin 4x \) with respect to \( x \), apply the chain rule:

• Differentiate the outer function (\( \sin \)) first: \[ \frac{d}{dx} (\sin 4x) = \cos(4x) \cdot \frac{d}{dx}(4x) \]
• Now take the derivative of the inside function, \( 4x \), with respect to \( x \), which is \( 4 \).

Putting it all together, you have:

\[ \frac{d}{dx} (\sin 4x) = \cos(4x) \cdot 4 = 4\cos(4x) \]

This means the derivative is 4cos4x. The result is positive, not negative, and the function is cosine (not sine) because the derivative of sine is cosine.