Question

find the derivative of y with respect to θ.
y = log 11 5θ
\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, \(\frac{dy}{d\theta}=\frac{1}{\ln 11}\cdot\frac{d}{d\theta}(\ln(5\theta))\).

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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:

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