Find the Derivative - d/dx (e^x)/(x^2)

Problem

d()/d(x)(ex)/(x2)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, u=ex and v=x2 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator u=ex with respect to x

d(ex)/d(x)=ex

3. Differentiate the denominator v=x2 with respect to x

d(x2)/d(x)=2*x

4. Substitute these components into the quotient rule formula.

d()/d(x)(ex)/(x2)=(x2*ex−ex*(2*x))/((x2)2)

5. Simplify the expression by factoring out common terms in the numerator and simplifying the denominator.

d()/d(x)(ex)/(x2)=(x*ex*(x−2))/(x4)

6. Reduce the fraction by canceling a factor of x from the numerator and denominator.

d()/d(x)(ex)/(x2)=(ex*(x−2))/(x3)

Final Answer

d()/d(x)(ex)/(x2)=(ex*(x−2))/(x3)

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