Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 34

$$ y' =\frac{1}{t}+\ln 5.$$

Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(\ln x)'=\dfrac{1}{x}$ Recall that $(e^x)'=e^x$ Since $ y=\ln(t5^t)$, then we have $$ y'=\frac{ 5^t+t 5^t \ln 5}{t5^t}=\frac{1+t \ln 5}{t }=\frac{1}{t}+\ln 5.$$