Find the Derivative - d/dx 1/2*(xe^(-1/2x))

Problem

d()/d(x)1/2*x*e(−1/2*x)

Solution

1. Apply the constant multiple rule to move the constant factor outside the derivative.

1/2(d(x)*e(−1/2*x))/d(x)

2. Apply the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=x and v=e(−1/2*x)

1/2*(xd(e(−1/2*x))/d(x)+e(−1/2*x)d(x)/d(x))

3. Apply the chain rule to differentiate e(−1/2*x) noting that the derivative of the exponent −1/2*x is −1/2

1/2*(x*(e(−1/2*x)⋅−1/2)+e(−1/2*x)⋅1)

4. Simplify the expression inside the parentheses by factoring out the common term e(−1/2*x)

1/2*e(−1/2*x)*(−1/2*x+1)

5. Distribute the constant 1/2 into the parentheses to reach the final form.

e(−1/2*x)*(−1/4*x+1/2)

Final Answer

d()/d(x)1/2*x*e(−1/2*x)=1/2*e(−1/2*x)−1/4*x*e(−1/2*x)

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