What is the derivative of 3sin(2x)?

Answer 1

We have a sinusoidal function f(x) = 3*sin(2x)

Apply the derivative shortcut rule for a sinusoidal function we know...

∂/∂x (sin(x)) = cos(x)

But we have 2x as the argument to the sine function, so we need to apply chain rule (or u-substitution)

let u = 2x

∂u/∂x = 2

Therefore...

∂/∂x [ 3sin(2x) ] = 3 ∂/∂x [ sin(2x) ] = 3 ∂/∂x [ sin(u) ]

This yields --> 3 * cos(u) * ∂u/∂x

Plugging the known substitution derivative and replacing u once again...

∂/∂x f(x) = 6cos(2x)