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

Problem

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

Solution

1. Identify the function as a product of two terms, u=e^x and v=ln(x) which requires the product rule.

2. Apply the product rule formula, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

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

4. Substitute these derivatives back into the product rule formula to get e^x1/x+ln(x)*e^x

5. Factor out the common term e^x to simplify the expression.

Final Answer

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

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