Question

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable. y equals (t )(t plus 2 )(t plus 6 ).

Answer 1

Answer:

$y' = [t^{(t+2)\cdot (t + 6)}]\cdot \left[(2\cdot t + 8)\cdot \ln t + \frac{t^{2}+8\cdot t + 12}{t} \right]$

Step-by-step explanation:

The function is:

$y = t^{(t+2)\cdot (t+6)}$

By using natural logarithmics and algebraic handling on both sides:

$\ln y = (t+2)\cdot (t+ 6)\cdot \ln t$

$\ln y = (t^{2}+8\cdot t + 12)\cdot \ln t$

The logarithmic differentiation is:

$\frac{1}{y}\cdot y' = (2\cdot t + 8)\cdot \ln t + \frac{t^{2}+8\cdot t + 12}{t}$

$y' = y\cdot \left[(2\cdot t + 8)\cdot \ln t + \frac{t^{2}+8\cdot t + 12}{t} \right]$

$y' = [t^{(t+2)\cdot (t + 6)}]\cdot \left[(2\cdot t + 8)\cdot \ln t + \frac{t^{2}+8\cdot t + 12}{t} \right]$