Find dy/dx y=e^(2x)sin(x)
The problem provided asks to compute the derivative of the function y with respect to the variable x, where y is defined as a composition of exponential and trigonometric functions, specifically y is equal to e raised to the power of 2x, multiplied by the sine of x. The notation dy/dx represents this derivative, which is looking for the rate at which y changes with a small change in x.
Apply the differentiation operator to both sides of the given function:
The derivative of
Proceed to differentiate the expression on the right-hand side.
Utilize the Product Rule for differentiation:
The derivative of
Apply the Chain Rule for differentiation, which states
Introduce a substitution, let
Differentiate using the Exponential Rule:
Substitute back
Perform the differentiation.
As
Apply the Power Rule:
Simplify the expression.
Multiply
Rearrange to place the constant
Combine like terms to obtain
Express the derivative of
Replace
Product Rule: When differentiating the product of two functions, the derivative is the first function times the derivative of the second function plus the second function times the derivative of the first function.
Chain Rule: Used to differentiate composite functions. If a variable is a function of another variable (e.g.,
Exponential Rule: The derivative of
Power Rule: When differentiating
Simplification: In calculus, it is often necessary to simplify expressions by combining like terms or rearranging terms to make the differentiation process clearer.
Notation: The notation