Answer:
[tex]\displaystyle \frac{dy}{d\theta} = \frac{1}{\theta -4} -e^\theta[/tex]
Step-by-step explanation:
We are given the function:
[tex]\displaystyle y = \ln \left( \theta - 4 \right) - e^\theta[/tex]
And we want to find dy/dθ.
Take the derivative of both sides with respect to θ:
[tex]\displaystyle \frac{dy}{d\theta} = \frac{d}{d\theta}\left[ \ln \left(\theta - 4\right) - e^\theta\right][/tex]
Rewrite:
[tex]\displaystyle \frac{dy}{d\theta} = \frac{d}{d\theta}\left[ \ln\left(\theta -4\right)\right] - \frac{d}{d\theta}\left[e^\theta\right][/tex]
Differentiate. The first differentiation requires the chain rule:
[tex]\displaystyle \begin{aligned}\frac{dy}{d\theta} & = \frac{1}{(\theta - 4)}\cdot \frac{d}{d\theta}\left[ \theta -4\right] - (e^\theta) \\ \\ & = \frac{1}{\theta -4} - e^\theta \end{aligned}[/tex]
In conclusion:
[tex]\displaystyle \frac{dy}{d\theta} = \frac{1}{\theta -4} -e^\theta[/tex]
