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

Question

Emily Ammerman · Accepted Answer

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

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)#

Cameron Alexander · Answer

undefined To differentiate $ y = 3x \sin(6x) $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function.
   - Outer function: $ u = 3x $ (multiplied by sin(6x))
   - Inner function: $ v = 6x $ (inside the sine function)

2. Compute the derivative of the outer function with respect to the inner function:
   - $ \frac{du}{dv} = 3 $

3. Compute the derivative of the inner function with respect to the variable:
   - $ \frac{dv}{dx} = 6 $

4. Apply the chain rule formula: $ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} $.
   - $ \frac{dy}{dx} = 3 \cdot 6 $
   - $ \frac{dy}{dx} = 18x \cos(6x) $

So, the derivative of $ y = 3x \sin(6x) $ is $ \frac{dy}{dx} = 18x \cos(6x) $.