Explanation:
To find the derivative of the function y = (x^3 * e^(x+5))^2x using logarithmic differentiation, we follow these steps:
Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln((x^3 * e^(x+5))^2x)
Apply the properties of logarithms to simplify the expression:
ln(y) = 2x * ln(x^3 * e^(x+5))
Expand the logarithm using the properties of logarithms:
ln(y) = 2x * (ln(x^3) + ln(e^(x+5)))
Simplify the logarithms:
ln(y) = 2x * (3ln(x) + (x+5))
Differentiate both sides of the equation with respect to x:
(d/dx) ln(y) = (d/dx) [2x * (3ln(x) + (x+5))]
Apply the chain rule and simplify:
(dy/y) = [2 * (3ln(x) + (x+5))] + [2x * (3/x) + 1]
Rearrange the equation to solve for dy/dx:
dy/dx = y * [2 * (3ln(x) + (x+5))] + [2x * (3/x) + 1]
Finally, we can substitute back y = (x^3 * e^(x+5))^2x into the equation to obtain the derivative in terms of x.