Question

find the derivative of y with respect to θ.
y = log₇8θ

dy/dθ = □

Explanation:

Step1: Use change - of - base formula

First, rewrite $y = \log_{7}(8\theta)$ using the change - of - base formula $\log_{a}u=\frac{\ln u}{\ln a}$. So $y=\frac{\ln(8\theta)}{\ln 7}$. Since $\ln 7$ is a constant, we can rewrite $y$ as $y=\frac{1}{\ln 7}\ln(8\theta)$.

Step2: Apply the chain - rule for differentiation

The derivative of $\ln(u)$ with respect to $\theta$ is $\frac{u'}{u}$. Here $u = 8\theta$, so $u'=8$. The derivative of $y$ with respect to $\theta$ is $\frac{dy}{d\theta}=\frac{1}{\ln 7}\cdot\frac{8}{8\theta}$.

Step3: Simplify the expression

$\frac{1}{\ln 7}\cdot\frac{8}{8\theta}=\frac{1}{\theta\ln 7}$.

Answer:

$\frac{1}{\theta\ln 7}$