Find the Derivative - d/dx 2xe^x

Problem

(d(2)*x*ex)/d(x)

Solution

1. Identify the rule needed for differentiation. Since the expression 2*x*ex is a product of two functions, 2*x and ex the product rule must be applied.

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) Let u=2*x and v=ex

3. Differentiate the individual components. The derivative of 2*x is 2 and the derivative of ex is ex

4. Substitute these values into the product rule formula:

(d(2)*x*ex)/d(x)=2*xd(ex)/d(x)+ex(d(2)*x)/d(x)

5. Calculate the derivatives:

(d(2)*x*ex)/d(x)=2*x(ex)+e(2)x

6. Simplify the expression by factoring out the common term 2*ex

(d(2)*x*ex)/d(x)=2*x*ex+2*ex

(d(2)*x*ex)/d(x)=2*ex*(x+1)

Final Answer

(d(2)*x*ex)/d(x)=2*ex*(x+1)

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