Find the Derivative - d/dx 2xe^(-x)

Problem

d()/d(x)*2*x*e(−x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, 2*x and e(−x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the variables for the product rule. Let u=2*x and v=e(−x)

3. Differentiate each part separately. The derivative of u is (d(2)*x)/d(x)=2 The derivative of v requires the chain rule: d(e(−x))/d(x)=e(−x)⋅(d(−)*x)/d(x)=−e(−x)

4. Apply the product rule formula by substituting the parts back in.

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

5. Simplify the expression by factoring out the common term 2*e(−x)

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

(d(2)*x*e(−x))/d(x)=2*e(−x)*(1−x)

Final Answer

(d(2)*x*e(−x))/d(x)=2*e(−x)*(1−x)

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