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

Problem

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

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the numerator u=ln(x) and the denominator v=e^x

3. Differentiate the individual components: d(ln(x))/d(x)=1/x and d(e^x)/d(x)=e^x

4. Substitute these values into the quotient rule formula.

(e^x1/x−ln(x)*e^x)/((e^x)^2)

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

(e^x*(1/x−ln(x)))/((e^x)^2)

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

(1/x−ln(x))/(e^x)

7. Find a common denominator for the numerator to remove the complex fraction.

(1−x*ln(x))/x/(e^x)

8. Finalize the expression by moving x to the denominator.

(1−x*ln(x))/(x*e^x)

Final Answer

d()/d(x)ln(x)/(e^x)=(1−x*ln(x))/(x*e^x)

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