Expert: Ahmed Salami - 11/13/2009

Question
Use Logarithmic differentiation to find the derivative of the function.
y= (rad x)(e^xsquared)(x^2+2)^10

I have gotten to
ln(y)= 1/2 ln(x) • 2x• 10 ln(x^2+1)
and I do not know if I am correct and if I am
I do not know what the next step would be.

Answer
Hi Patty,
y = √x . e^x² . (x² + 2)^10
Taking logarithms,
ln y = ln[√x . e^x² . (x² + 2)^10]
ln y = ln√x + ln(e^x²) + ln(x² + 2)^10
ln y = (1/2)ln x + x²ln(e) + 10ln(x² + 2)
ln y = (1/2)ln x + x² + 10ln(x² + 2)
Differentiating each term,
(1/y)dy/dx = (1/2)(1/x) + 2x + 10[1/(x² + 2)].2x
(1/y)dy/dx = (1/2x)  +  2x  +  20x/(x² + 2)
dy/dx = y . [(1/2x)  +  2x  +  20x/(x² + 2)]
OR
dy/dx = [√x . e^x² . (x² + 2)^10] . [(1/2x)  +  2x  +  20x/(x² + 2)]

Regards