Find the Derivative - d/dx log of 2x

Problem

d((log_)(2*x))/d(x)

Solution

1. Identify the function as a common logarithm, which is typically base 10, denoted as (log_10)(2*x)

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

(log_10)(2*x)=ln(2*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((log_)(2*x))/d(x)=1/ln(10)⋅1/(2*x)⋅(d(2)*x)/d(x)

4. Differentiate the inner function 2*x with respect to x

(d(2)*x)/d(x)=2

5. Substitute the derivative of the inner function back into the expression.

d((log_)(2*x))/d(x)=1/ln(10)⋅1/(2*x)⋅2

6. Simplify the expression by canceling the constant factor 2.

d((log_)(2*x))/d(x)=1/(x*ln(10))

Final Answer

d((log_)(2*x))/d(x)=1/(x*ln(10))

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