What is the derivative of y=(e^(2x))(sin(3x))
This question is slightly complex and requires you to use multiple differentiation rules. Since we are differentiating a product of two functions, the first rule that we have to use is the product rule: multiply the derivative of the first function (e^(2x)) by the second function, then add the derivative of the second function (sin(3x)) times the first function.
e^(2x) is in the form of a function within a function. The derivative of e^(2x) is found by using the chain rule: multiply the derivative of the function g(x)=2x by the derivative of e^g(x). This gives us 2e^(2x). Similarly, sin(3x) is also a function within a function and we must use the chain rule again, and multiply the derivative of the function h(x)=3x by the derivative of sin(h(x)), which gives us 3cos(3x).
After some simple rearranging of terms, we result in the answer to our question. dy/dx=e^(2x)(2sin(3x) + 3cos(3x))
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