Question

How do you use the chain rule to differentiate `y=3xsin(6x)`?

Answer 1

`dy/dx=18xcos(6x)+3sin(6x)`

Explanation:

The chain rule states that after taking the derivative of a function with respect to another function, we must multiply the result by the derivative of the inside function. In math terms:

`f(g(x))'=f'(g(x))*g'(x)`

Let's apply the rule to the question:

`y=3xsin(6x)`

To find `dy/dx`, we first have to apply the product rule:

`dy/dx=3sin(6x)+3x*d/dx(sin(6x))`

To find the derivative of `sin(6x)`, we first take the derivative of the outside function:

`d/(d(6x))sin(6x)=cos(6x)`

then multiply by the result by the derivative of the inside function:

`d/dx6x=6`

So the expression becomes:

`rArrdy/dx=3sin(6x)+3x*cos(6x)*6`

`rArrdy/dx=18xcos(6x)+3sin(6x)`