Question

find the derivative of y with respect to x.
y = log 3 e^{2x}
\frac{dy}{dx} = \square

Explanation:

Step1: Use the change - of - base formula

The change - of - base formula for logarithms is \(\log_{a}b=\frac{\ln b}{\ln a}\). So, \(y = \log_{3}e^{2x}=\frac{\ln(e^{2x})}{\ln 3}\). Since \(\ln(e^{u})=u\) (by the property of logarithms \(\ln(e^{u}) = u\ln e\) and \(\ln e = 1\)), then \(y=\frac{2x}{\ln 3}\).

Step2: Differentiate using the power rule

The power rule for differentiation is \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\). For \(y=\frac{2x}{\ln 3}=\frac{2}{\ln 3}x\) (where \(a = \frac{2}{\ln 3}\) and \(n = 1\)), then \(\frac{dy}{dx}=\frac{2}{\ln 3}\times1\).

Answer:

\(\frac{2}{\ln 3}\)