Find the Derivative - d/dx y = log of 6x

Problem

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

Solution

1. Identify the function as a common logarithm, which is typically 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)

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

3. Apply the chain rule and the derivative rule for natural logarithms, where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

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

4. Differentiate the inner function 6*x with respect to x

(d(6)*x)/d(x)=6

5. Simplify the resulting expression by canceling the constant factors.

1/ln(10)⋅1/(6*x)⋅6=1/(x*ln(10))

Final Answer

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

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