Question

For any positive constant \(k,\) the derivative of \(\ln (k x)\) is 1\(/ x\) (a) by using the Chain Rule. (b) by using a property of logarithms and differentiating.

Short answer

The derivative of \(\ln(kx)\) is \(\frac{1}{x}\), whether computed using the chain rule or properties of logarithms for differentiation.

Step-by-step solution

Derivative of \(\ln (k x)\) using the Chain Rule

Applying the chain rule to find the derivative of a composition of two functions, we get that the derivative of \(\ln(kx)\) where \(k\) is constant and \(x\) is the variable, is \(\frac{1}{kx} * k = \frac{1}{x}\).

Derivative of \(\ln (k x)\) using properties of logarithms and differentiating

By using the property of logarithms that says \(\ln(ab)=\ln(a)+\ln(b)\), we can rewrite \(\ln(kx)\) as \(\ln(k)+\ln(x)\). The derivative of \(\ln(k)\) is 0 because \(k\) is a constant, and the derivative of \(\ln(x)\) is \(\frac{1}{x}\). So, the derivative of \(\ln(kx)\) is \(\frac{1}{x}\) + 0 = \(\frac{1}{x}\).