Find the Derivative - d/dx log base 8 of xe^x

Problem

d((log_8)(x*ex))/d(x)

Solution

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

(log_8)(x*ex)=ln(x*ex)/ln(8)

2. Use logarithm properties to simplify the numerator before differentiating.

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

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

3. Rewrite the expression to be differentiated by pulling out the constant factor.

d()/d(x)(ln(x)+x)/ln(8)=1/ln(8)d(ln(x)+x)/d(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(8)*(1/x+1)=1/ln(8)*((1+x)/x)

Final Answer

d((log_8)(x*ex))/d(x)=(x+1)/(x*ln(8))

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