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 \( 4 \).
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.
