find y' and y''

Question

y = (square root x) (ln(x))

please show all work and then explain in comments, thank you!

Answer 1

To answer this question, we have to use the product rule. So the first derivative of sqrt(x)*ln(x) is:

y'=x^1/2/x + 1/2ln(x)/x^1/2

The product rule is given by fg'+f'g

we can stand to rewrite y':

y'=x^-1/2+1/2*ln(x)*x^-1/2

Now we find the second derivative:

y''=-1/2x^(-3/2) + 1/2(-1/2ln(x)*x^(-3/2)+(x^(-1/2))/x) <- Product rule yet again.

y''=-1/2*x^(-3/2)-1/4*ln(x)*x^(-3/2)+1/2*x^(-3/2)

That last part is the simplified version of the second derivative. I hope this helped.