Find the Derivative - d/dx y = log of 4x

Problem

d()/d(x)*(log_)(4*x)

Solution

1. Identify the function as a common logarithm, which is typically assumed to be base 10, denoted as (log_10)(4*x)

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

(log_10)(4*x)=ln(4*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(4*x)/ln(10)=1/ln(10)⋅1/(4*x)⋅(d(4)*x)/d(x)

4. Differentiate the inner function 4*x which results in 4

1/ln(10)⋅1/(4*x)⋅4

5. Simplify the expression by canceling the constant factor 4.

1/(x*ln(10))

Final Answer

d((log_)(4*x))/d(x)=1/(x*ln(10))

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