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

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two functions, u=ln(x) and v=ex apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator and the denominator separately.

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

d(ex)/d(x)=ex

3. Substitute these derivatives into the quotient rule formula.

d()/d(x)ln(x)/(ex)=(ex⋅1/x−ln(x)⋅ex)/((ex)2)

4. Factor out the common term ex from the numerator to simplify the expression.

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

5. Simplify the fraction by canceling ex and combining the terms in the numerator.

d()/d(x)ln(x)/(ex)=(1/x−ln(x))/(ex)

d()/d(x)ln(x)/(ex)=(1−x*ln(x))/(x*ex)

Final Answer

d()/d(x)ln(x)/(ex)=(1−x*ln(x))/(x*ex)

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