To find the derivative of the function y = sin(3x) * sin(6x), we apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Start with the original function:
y = sin(3x) * sin(6x)
Identify two functions involved:
f(x) = sin(3x)
g(x) = sin(6x)
We have to find the derivatives for these functions:
f'(x) = cos(3x) * 3
g'(x) = cos(6x) * 6
Now we can apply the product rule:
y' = f(x)g'(x) + g(x)f'(x)
This equation translates to:
y' = sin(3x)[cos(6x)*6] + sin(6x)[cos(3x)*3]
Simplify the resulting equation:
y' = 6sin(3x)cos(6x) + 3sin(6x)cos(3x)
It states that the derivative of the function y = sin(3x) * sin(6x) is 6sin(3x)cos(6x) + 3sin(6x)cos(3x).
sin(3x) * sin(6x) = 6sin(3x)cos(6x) + 3sin(6x)cos(3x)