# Easy derivative question

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## Homework Statement

derive this function

f(x) = (2x + x^3) / sqrt(x)

## The Attempt at a Solution

how can i derive this without using quotient rule? my prof is asking to do so without using it.

I

Geofleur
Gold Member
Can you rewrite it in a way that would allow you to use the product rule?

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

derive this function

f(x) = (2x + x^3) / sqrt(x)

## The Attempt at a Solution

how can i derive this without using quotient rule? my prof is asking to do so without using it.

I
Write ##\ \sqrt{x} \ ## as ##\displaystyle \ x^{1/2} \ ## .

By the way, the word in English is differentiate, not derive.

Can this be done without product / quotient rule?

Just to remind myself of how quotient rule works...

I got to this point after using quotient rule:
((2x + 7x^3) / (2(x^1/2))) / x

How do i simplify this? Sorry I don't know how to use latex code.

Geofleur
Gold Member
OK, here's a hint for how to do it without the product rule either: ##x^m / x^n = x^{(m-n)} ##.

Okay using that I got something like x^5/2 + x^1/2. It doesn't seem correct, how would i account for the binomial on the numerator? would I write another expression for x such as x^h?
\
edit: I just misunderstood the question... I am allowed to use both the quotient and product rule.

Mark44
Mentor
Okay using that I got something like x^5/2 + x^1/2. It doesn't seem correct, how would i account for the binomial on the numerator?
It's not correct, but you're not too far off. And you should end up with two terms, so I don't understand what you're asking about accounting for the binomial.
Arnoldjavs3 said:
would I write another expression for x such as x^h?
What do you get if you carry out the division below?
(2x + x^3) / sqrt(x)
Arnoldjavs3 said:
edit: I just misunderstood the question... I am allowed to use both the quotient and product rule.

alright... I got it.

I simply just subtract the exponents from the numerator and denominator so it became (2 + 5x^2) / 2x^1/2

Mark44
Mentor
alright... I got it.

I simply just subtract the exponents from the numerator and denominator so it became (2 + 5x^2) / 2x^1/2
No, that's not even close. There's some very basic algebra that you need to review. If you don't, you absolutely won't be able to do calculus.

##\frac{(a + b)} c = (a + b) \cdot \frac 1 c = a \cdot \frac 1 c + b \cdot \frac 1 c## . Use the distributive property to multiply each term of a + b by 1/c. Can you apply this idea to your problem, ##\frac{2x + x^3}{\sqrt{x}}##?