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