Find the Derivative - d/dx f(x) = log base 5 of xe^x

Problem

d()/d(x)*(log_5)(x*e^x)

Solution

1. Apply the change of base formula to rewrite the logarithm in terms of the natural logarithm.

(log_5)(x*e^x)=ln(x*e^x)/ln(5)

2. Use logarithm properties to simplify the numerator by splitting the product into a sum.

ln(x*e^x)=ln(x)+ln(e^x)

3. Simplify the expression further using the property ln(e^x)=x

(ln(x)+x)/ln(5)

4. Apply the constant multiple rule for differentiation, pulling out the constant factor 1/ln(5)

d()/d(x)(ln(x)+x)/ln(5)=1/ln(5)d(ln(x)+x)/d(x)

5. Differentiate the sum using the power rule and the derivative of the natural logarithm.

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

d(x)/d(x)=1

6. Combine the results into the final derivative expression.

1/ln(5)*(1/x+1)

Final Answer

d()/d(x)*(log_5)(x*e^x)=(1+x)/(x*ln(5))

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