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

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, u=ex and v=ln(x) use the quotient rule.

2. Apply the quotient rule formula, which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=ex to get d(ex)/d(x)=ex

4. Differentiate the denominator v=ln(x) to get d(ln(x))/d(x)=1/x

5. Substitute these derivatives back into the quotient rule formula.

(ln(x)⋅ex−ex⋅1/x)/((ln(x))2)

6. Simplify the expression by factoring out ex from the numerator and finding a common denominator for the terms in the numerator.

(ex*(ln(x)−1/x))/ln2(x)

7. Rewrite the expression to eliminate the fraction within the numerator by multiplying the top and bottom by x

(ex*(x*ln(x)−1))/(x*ln2(x))

Final Answer

d()/d(x)(ex)/ln(x)=(ex*(x*ln(x)−1))/(x*ln2(x))

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