Find 𝒅𝒚𝒅𝒙 at x=2 if y = 𝐭𝐚𝐧−𝟏(𝒙)+𝒆−𝒙𝑺𝒊𝒏𝟐𝒙

Sure, let's find the derivative of the function y = tan^(-1)(x) + e^(-x)sin(2x) at x=2.

Step 1: Differentiate each term separately.

The derivative of tan^(-1)(x) is 1/(1+x^2).

The derivative of e^(-x)sin(2x) can be found using the product rule and the chain rule. The product rule states that the derivative of two functions multiplied together is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

So, let's find the derivative of e^(-x) and sin(2x) separately first.

The derivative of e^(-x) is -e^(-x).

The derivative of sin(2x) is 2cos(2x).

Now, apply the product rule:

The derivative of e^(-x)sin(2x) is (-e^(-x))sin(2x) + e^(-x)(2cos(2x)).

Step 2: Evaluate the derivative at x=2.

Substitute x=2 into the derivative:

dy/dx at x=2 = 1/(1+(2)^2) + (-e^(-2))sin(4) + e^(-2)(2cos(4)).

Step 3: Simplify the expression.

dy/dx at x=2 = 1/5 - e^(-2)sin(4) + 2e^(-2)cos(4).

This is the derivative of the function at x=2.

Sure, let's find the derivative of the function y = tan^(-1)(x) + e^(-x)sin(2x) at x=2.

Step 1: Differentiate each term separately.

The derivative of tan^(-1)(x) is 1/(1+x^2).

The derivative of e^(-x)sin(2x) requires the product rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.

So, let's set u = e^(-x) and v = sin(2x).

The derivative of u, u', is -e^(-x).

The derivative of v, v', is 2cos(2x).

Applying the product rule, we get:

u'v + uv' = -e^(-x)sin(2x) + 2e^(-x)cos(2x).

Step 2: Add the derivatives together.

dy/dx = 1/(1+x^2) - e^(-x)sin(2x) + 2e^(-x)cos(2x).

Step 3: Substitute x = 2 into the derivative.

dy/dx = 1/(1+2^2) - e^(-2)sin(4) + 2e^(-2)cos(4).

This is the derivative of the function at x = 2.