Find the Derivative - d/dx x/(e^x)

Problem

d()/d(x)x/(e^x)

Solution

1. Identify the rule needed for the derivative of a quotient of two functions, which is the quotient rule.

2. Apply the formula for the quotient rule, d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2) where u=x and v=e^x

3. Calculate the individual derivatives d(x)/d(x)=1 and d(e^x)/d(x)=e^x

4. Substitute these values into the quotient rule formula.

d()/d(x)x/(e^x)=(e(1)^x−x(e^x))/((e^x)^2)

5. Factor out the common term e^x from the numerator.

d()/d(x)x/(e^x)=(e^x*(1−x))/((e^x)^2)

6. Simplify the expression by canceling e^x from the numerator and denominator.

d()/d(x)x/(e^x)=(1−x)/(e^x)

Final Answer

d()/d(x)x/(e^x)=(1−x)/(e^x)

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