Find the Derivative - d/dx log base 5 of xe^x

Problem

d()/d(x)*(log_5)(x*ex)

Solution

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

(log_5)(x*ex)=ln(x*ex)/ln(5)

2. Use logarithm properties to simplify the numerator by splitting the product and the exponent.

ln(x*ex)=ln(x)+ln(ex)

ln(x*ex)=ln(x)+x

3. Rewrite the expression to be differentiated using the constant multiple rule.

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

4. Differentiate the terms inside the parentheses using the sum rule.

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

d(x)/d(x)=1

5. Combine the results and simplify the expression into a single fraction.

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

Final Answer

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

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