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

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the function is a product of e^x and ln(x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the function to u and v Let u=e^x and v=ln(x)

3. Differentiate each part individually. The derivative of e^x is e^x and the derivative of ln(x) is 1/x

4. Apply the product rule formula by substituting the functions and their derivatives.

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

5. Substitute the derivatives into the expression.

(d(e^x)*ln(x))/d(x)=e^x1/x+ln(x)*e^x

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

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

Final Answer

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

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