How to get a derivative of a function without using the quotient, power, or chain rule?

Question

How to compute a derivative of

G(x)= (x^2 + 2x -1) / (sqrt(x))?

From what I'm understanding, I have to break it up so it looks like

x^2/sqrt(x) + 2x/sqrt(x) - 1/sqrt(x)

And after simplifying it would look like x*sqrt(x) +2*sqrt(x) - sqrt(x)/x

Which doesn't look right and doesn't come close to what the actual derivative is if I was to use the quotient rule. What am I doing wrong here?

Answer 1

Hey Eleanor!

Just because the answers you get from one method vs another look different, does not mean that they are not equal.

Here you have a function (x²+2x-1)/x^(1/2)

I wrote the square root as an exponent because it helps me when simplifying. If I want to break up the function and take the derivative of each term as I go, having it in this form makes it easier for me.

At this point if you used the quotient rule, you would get ((x^(1/2)*(2x +2)) - (1/2)x^(-1/2)*(x²+2x-1))/(x^(1/2))² which can then be algebraically simplified.

Instead we will break apart the numerator of the fraction and take the derivative of each term individually.

g(x)= x²/x^(1/2) + 2x/x^(1/2) - 1/x^(1/2)

Using exponent rules, we can simplify this further before taking the derivative

g(x) = x^(3/2) + 2x^(1/2) - x^(-1/2)

Now we take g'(x) and get

g'(x) = (3/2)*x^(1/2) + 2*(1/2)*x^(-1/2) - (-1/2)*x^(-3/2)

g'(x) = (3/2)*sqrt(x) + 1/sqrt(x) +1/(2*sqrt(x³))

Answer 2

G(x)= (x^2 + 2x -1) / x^.5

= (x^4+2x^3-x^2+2x^3+4x^2-2x-x^2-2x+1)/x

=(x^4+4x^3+2x^2-4x+1)/x

=x^3+4x^2+2x-4+1/x

=x^3+4x^2+2x+1/x=4

x=.4

Answer 3

Thanks for your question, Eleanor. Two things:

1. You can break the fraction into three terms like you did but you do not have to.
2. You have only changed what it looks like and have not taken the derivative yet.

If you want to find the derivative without the rules that make it easy, then you must use the definition of the derivative: G'(x) = lim h->0 [G(a + h) - G(a)]/h

Or the alternative definition: G'(x) = lim x->a [G(x) - G(a)] / [x - a]

Use either the given G(x) or the one that you rewrote, substitute each part, and find the limit. Then you can check by the chain rule, product rule, and quotient rule.