what is the derivitive of this function using the logarithmic rules?

Question

y= sqrt( x) * e^(x^2) * (x^2+1)^10

Answer 1

lny = x^1/2 +x^2 +10 ln(x^2+1)

y´/y = 1/2 x^-1/2 + 2x +20x/(x^2+1)

so y´= [ 1/2x^-1/2 +2x+20x/(x^2+1)]*x^1/2*e^(x^2)*(x^2+1)^10

Answer 2

ln y = (1/2) ln x + x^2 + 10 [ln (x^2 + 1)], therefore :

(dy/dx) / y = (1/2x) + 2 x + 10 [2 x / (x^2 + 1)] = (1/2x) + 2 x + 20 x / (x^2 + 1).

So, dy/dx = [(1/2x) + 2 x + 20 x / (x^2 + 1)] {(1/2) ln x + x^2 + 10 [ln (x^2 + 1)]}.

Live long and prosper.

P.S. Note that, unfortunately, the preceding responder, 'santmann2002,' forgot to take the natural logarithm of sqrt (x) [or x^(1/2)], before doing any differentiation.