Question

if ( y = sin(3x) ), then ( \frac{dy}{dx} = )

a ( -3cos(3x) )
b ( -cos(3x) )
c ( -\frac{1}{3}cos(3x) )
d ( cos(3x) )

3 cos (3x)

Explanation:

Step1: Apply the chain rule

Let \(u = 3x\), then \(y=\sin(u)\). The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
For \(y = \sin(u)\), \(\frac{dy}{du}=\cos(u)\).
For \(u = 3x\), \(\frac{du}{dx}=3\).

Step2: Substitute back

Substitute \(u = 3x\) and the derivatives into the chain - rule formula: \(\frac{dy}{dx}=\cos(3x)\cdot3 = 3\cos(3x)\).

Answer:

E. \(3\cos(3x)\)