Problem

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.

y=e2xsin(x)

Answer

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Solution:

Step:1

Apply the differentiation operator to both sides of the given function: ddx(y)=ddx(e2xsin(x)).

Step:2

The derivative of y with respect to x is denoted as dydx.

Step:3

Proceed to differentiate the expression on the right-hand side.

Step:3.1

Utilize the Product Rule for differentiation: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)], where f(x)=e2x and g(x)=sin(x). Thus, we have e2xddx[sin(x)]+sin(x)ddx[e2x].

Step:3.2

The derivative of sin(x) with respect to x is cos(x). This gives us e2xcos(x)+sin(x)ddx[e2x].

Step:3.3

Apply the Chain Rule for differentiation, which states ddx[f(g(x))]=f(g(x))g(x), where f(x)=ex and g(x)=2x.

Step:3.3.1

Introduce a substitution, let u=2x. The expression becomes e2xcos(x)+sin(x)(ddu[eu]ddx[2x]).

Step:3.3.2

Differentiate using the Exponential Rule: ddu[eu]=euln(e), since a=e. This simplifies to e2xcos(x)+sin(x)(euddx[2x]).

Step:3.3.3

Substitute back u with 2x to get e2xcos(x)+sin(x)(e2xddx[2x]).

Step:3.4

Perform the differentiation.

Step:3.4.1

As 2 is a constant, the derivative of 2x with respect to x is 2ddx[x]. This gives us e2xcos(x)+sin(x)(e2x(2ddx[x])).

Step:3.4.2

Apply the Power Rule: ddx[xn]=nxn1, where n=1. This simplifies to e2xcos(x)+sin(x)(e2x(21)).

Step:3.4.3

Simplify the expression.

Step:3.4.3.1

Multiply 2 by 1 to get e2xcos(x)+sin(x)(e2x2).

Step:3.4.3.2

Rearrange to place the constant 2 in front of e2x, resulting in e2xcos(x)+sin(x)(2e2x).

Step:3.4.3.3

Combine like terms to obtain e2xcos(x)+2e2xsin(x).

Step:4

Express the derivative of y as equal to the simplified right-hand side: dydx=e2xcos(x)+2e2xsin(x).

Step:5

Replace dydx with y to denote the derivative of y with respect to x: y=e2xcos(x)+2e2xsin(x).

Knowledge Notes:

  1. 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.

  2. Chain Rule: Used to differentiate composite functions. If a variable is a function of another variable (e.g., u=g(x)), then the derivative of a function of that variable (e.g., f(u)) is the derivative of the outer function with respect to the inner function times the derivative of the inner function with respect to x.

  3. Exponential Rule: The derivative of eu with respect to u is eu itself, and if u is a function of x, then the derivative involves the chain rule.

  4. Power Rule: When differentiating xn with respect to x, the result is nxn1.

  5. Simplification: In calculus, it is often necessary to simplify expressions by combining like terms or rearranging terms to make the differentiation process clearer.

  6. Notation: The notation dydx and y are both used to denote the derivative of y with respect to x.

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