Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am suppose to differentiate $(x^2 +4x +3)/ \sqrt{x}$ I know that a square root is equal to $x^{1/2}$ but I still am not able to properly differentiate this problem. I ended up with $(2x+4)/ (1/2)x^{1/2})$ which I know is wrong.

share|cite|improve this question
You seem to be trying to differentiate the numerator and the denominator separately; that does not work. You have to either use the Quotient Rule or, if you don't know it yet, first simplify the expression using algebra:$$\frac{x^2+4x+3}{\sqrt{x}} = \frac{x^2+4x+3}{x^{1/2}} = \frac{x^2}{x^{1/2}}+\frac{4x}{x^{1/2}} + \frac{3}{x^{1/2}} = x^{3/2}+4x^{1/2}+3x^{-1/2}$$and then differentiate. – Arturo Magidin Sep 17 '11 at 21:05
Hint: $\frac{a+b+c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$. – André Nicolas Sep 17 '11 at 21:07
I am suppose to be learning the sum, difference and power rule. I have not yet encountered the sum or difference rule and I have no idea how to use them. – user138246 Sep 17 '11 at 23:06
The "sum rule" just says that $\frac{d}{dx}(f+g) = \frac{d}{dx}f + \frac{d}{dx}g$ (the derivative of a sum is the sum of the derivatives), and the "difference rule" just says that $\frac{d}{dx}(f-g) = \frac{d}{dx}f - \frac{d}{dx}g$, the derivative of the difference is the difference of the derivatives. In fact, you've used them alread, when you were trying to take the derivative of a polynomial by doing it for each term and then adding. This works for sums/difference, but not for products or quotients. – Arturo Magidin Sep 17 '11 at 23:12
@Jordan: The power rule works for any exponent, so long as the base is just the variable by itself, and the exponent is constant. $\displaystyle \frac{d}{dx}x^n = nx^{n-1}$. So for $x^{-1/2}$, you get $$\frac{d}{dx}x^{-1/2} = -\frac{1}{2}x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}}.$$Just be careful with the subtraction in the exponent. For $3x^{-1/2}$, you have: $$\frac{d}{dx}\left(3x^{-1/2}\right) = 3\frac{d}{dx}x^{-1/2} = 3\left(-\frac{1}{2}\right)x^{-\frac{3}{2}}= -\frac{3}{2}x^{-3/2}.$$The fact that the coefficient equals the exponent here is coincidence, don't read anything into it. – Arturo Magidin Sep 17 '11 at 23:46
up vote 5 down vote accepted

Dividing through, your expression equals $$ x^{3/2}+4x^{1/2}+3x^{-1/2}. $$ Now you can use the power rule.

share|cite|improve this answer

You can also exercise your quotient rule: $\left( \dfrac{f}{g}\right)^'=\dfrac{f'g-gf'}{g^2}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.