# Possible mistake during differentiation? Please check

• Dominathan
Let's take a look at what you have so far:You are correct in that the quotient rule requires you to divide by the square of the denominator. However, in this case, you are dividing by 2, which is the square of 1.

#### Dominathan

For a function f(x), I have to determine intervals of increase/decrease, find local max(s)/min(s), and find intervals of concavity. The first thing I'm doing in this is to write out f'(x) and f''(x).

f(x) = $ln(x)/\sqrt{x}$

For f'(x), I used the quotient rule and received f'(x) = (($\frac{1}{x}$$\sqrt{x}$)-($\frac{-\sqrt{x}}{2}$ln(x))) / 2

However, I plugged f(x) into wolfram alpha and it gave me: $\frac{2-ln(x)}{2x^{3/2}}$

I don't understand the difference? I thought I had done this correctly but apparently not? Wolfram alpha used the product rule. Is there some kind of algebraic gymnastics I'm forgetting about? I really want to understand where my error was made, not just which is the correct answer. Thanks!

In using the quotient rule you are not differentiating $\sqrt{x}$ corretly and the quotient rule requires you to divide by the square of the denominator.

Dominathan said:
f(x) = $ln(x)/\sqrt{x}$

For f'(x), I used the quotient rule and received f'(x) = (($\frac{1}{x}$$\sqrt{x}$)-($\frac{-\sqrt{x}}{2}$ln(x))) / 2

I see a couple of issues here:

1) How did you get $\frac{-\sqrt{x}}{2}$ in the numerator? What is the derivative of $\sqrt{x}$ ?

2) How did you get 2 in the denominator? $(\sqrt{x})^{2} =$ ?

Using quotient rule, you should get $\frac{\frac{\sqrt{x}}{x} - \frac{\ln{x}}{2 \sqrt{x}}}{x}$ which simplifies to what you got from WA. It looks like you messed up on the derivative of $\sqrt{x}$ and on the bottom of the quotient rule. http://en.wikipedia.org/wiki/Quotient_rule

gb7nash said:
I see a couple of issues here:

1) How did you get $\frac{-\sqrt{x}}{2}$ in the numerator? What is the derivative of $\sqrt{x}$ ?

2) How did you get 2 in the denominator? $(\sqrt{x})^{2} =$ ?

1.
1. $\sqrt{x}$ = x$^{1/2}$
2. Using the power rule I bring the 1/2 out as a coefficient, and subtract one from the numerator : $\frac{1}{2}$$x^{-1/2}$
3. I simplified to : $\frac{-\sqrt{x}}{2}$