Find the Derivative - d/dx log of 6x

Problem

d((log_)(6*x))/d(x)

Solution

1. Identify the function as a common logarithm, which is assumed to be base 10, denoted as (log_10)(6*x)

2. Apply the change of base formula to rewrite the expression in terms of the natural logarithm, ln(x) using the identity (log_b)(u)=ln(u)/ln(b)

(log_10)(6*x)=ln(6*x)/ln(10)

3. Apply the constant multiple rule by pulling the constant factor 1/ln(10) out of the derivative.

d((log_)(6*x))/d(x)=1/ln(10)d(ln(6*x))/d(x)

4. Apply the chain rule to differentiate ln(6*x) where the derivative of ln(u) is 1/u⋅d(u)/d(x)

d(ln(6*x))/d(x)=1/(6*x)⋅6

5. Simplify the expression by canceling the factor of 6 in the numerator and denominator.

1/(6*x)⋅6=1/x

6. Combine the results to find the final derivative.

d((log_)(6*x))/d(x)=1/(x*ln(10))

Final Answer

d((log_)(6*x))/d(x)=1/(x*ln(10))

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