Find the Derivative - d/d@VAR f(x)=( natural log of x)/(e^x)

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two functions, u=ln(x) and v=e^x apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Differentiate the numerator and the denominator separately.

d(ln(x))/d(x)=1/x

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

3. Substitute these derivatives into the quotient rule formula.

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

4. Factor out the common term e^x from the numerator to simplify the expression.

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

5. Simplify the fraction by canceling e^x and combining the terms in the numerator.

d()/d(x)ln(x)/(e^x)=(1/x−ln(x))/(e^x)

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

Final Answer

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

Want more problems? Check here!