find the derivative of y with respect to θ.\ny = log 11 5θ\n\\frac{dy}{dθ} = \\square

Explanation:

Step1: Use the change - of - base formula

The formula for changing the base of a logarithm is (\log_{a}u=\frac{\ln u}{\ln a}). So, (y = \log_{11}(5\theta)=\frac{\ln(5\theta)}{\ln 11}). Since (\frac{1}{\ln 11}) is a constant, (y=\frac{1}{\ln 11}\cdot\ln(5\theta)).

Step2: Apply the chain rule

The derivative of (\ln u) with respect to (x) is (\frac{u'}{u}) (by the chain rule, where (u = 5\theta) and (u'=5)). So, (\frac{dy}{d\theta}=\frac{1}{\ln 11}\cdot\frac{d}{d\theta}(\ln(5\theta))). [ \begin{align*} \frac{dy}{d\theta}&=\frac{1}{\ln 11}\cdot\frac{5}{5\theta}\ \end{align*} ]

Step3: Simplify the expression

Cancel out the common factor of (5) in the numerator and denominator. (\frac{dy}{d\theta}=\frac{1}{\theta\ln 11})

Answer:

(\frac{1}{\theta\ln 11})