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

Problem

d()/d(x)x/(ex)

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))/(v2) where u=x and v=ex

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

4. Substitute these values into the quotient rule formula.

d()/d(x)x/(ex)=(e(1)x−x(ex))/((ex)2)

5. Factor out the common term ex from the numerator.

d()/d(x)x/(ex)=(ex*(1−x))/((ex)2)

6. Simplify the expression by canceling ex from the numerator and denominator.

d()/d(x)x/(ex)=(1−x)/(ex)

Final Answer

d()/d(x)x/(ex)=(1−x)/(ex)

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