Find the Derivative - d/dx log base 6 of x

Problem

d((log_6)(x))/d(x)

Solution

1. Identify the function as a logarithm with a base other than e

2. Apply the change of base formula to rewrite the logarithm in terms of the natural logarithm, ln(x)

(log_6)(x)=ln(x)/ln(6)

3. Apply the constant multiple rule for differentiation, treating 1/ln(6) as a constant.

d((log_6)(x))/d(x)=1/ln(6)d(ln(x))/d(x)

4. Differentiate the natural logarithm using the rule d(ln(x))/d(x)=1/x

d((log_6)(x))/d(x)=1/ln(6)⋅1/x

5. Simplify the expression by combining the terms into a single fraction.

d((log_6)(x))/d(x)=1/(x*ln(6))

Final Answer

d((log_6)(x))/d(x)=1/(x*ln(6))

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