Mind blown - probably missing algebra concept - cannot simplify derivative

frank789

Junior Member
hi all!

sorry may have been at this too long but ive been blanking on this one for a while now, been coming at it different ways and I cannot work it down!

f(x) = x^2/sqrt(x^2-1)

so what i did for f'(x)

f(x) = x^2 * 1/sqrt(x^2-1)

product and chain rule to:

f'(x) = 2x/sqrt(x^2-1) - x^3/(x^2-1)^(3/2)

I cannot seem for the life of me to algebra this little rascal down a single term and I feel like I should be able to as solving in this state for the critical points is proving....lets use challenging.

Would really appreciate a point in the right direction!

Subhotosh Khan

Super Moderator
Staff member
hi all!

sorry may have been at this too long but ive been blanking on this one for a while now, been coming at it different ways and I cannot work it down!

f(x) = x^2/sqrt(x^2-1)

so what i did for f'(x)

f(x) = x^2 * 1/sqrt(x^2-1)

product and chain rule to:

f'(x) = 2x/sqrt(x^2-1) - x^3/(x^2-1)^(3/2)

I cannot seem for the life of me to algebra this little rascal down a single term and I feel like I should be able to as solving in this state for the critical points is proving....lets use challenging.

Would really appreciate a point in the right direction!
Much easier to apply quotient rule.

$$\displaystyle \frac{d}{dx} \left[\dfrac{g(x)}{h(x}\right] = \ \dfrac{g'(x)*h(x) - g(x)*h'(x)}{[h(x)]^2}$$

g(x) = x^2 → g'(x) = 2x &

h(x) = (x^2 - 1)^(1/2) → h'(x) = x * (x^2 - 1)^(-1/2)

Now continue.....

frank789

Junior Member
ok I went through product rule

i got

2x(x^2-1)^(1/2) - (x^3)(x^2-1)^(-1/2)
----------------------------------------------
x^2 - 1

then I dived each term by the x^2 - 1 to get

(2x)(x^2 - 1)^(-1/2) - (x^3)(x^2 - 1)^(-3/2)

which if im not mistaken is the same answer I reached before. as im typing this the first few sips of coffee #2 seem to have done the trick

factor out the largest power of (x^2 - 1), which is (-3/2), along with an x to give

x(x^2 - 1)^(-3/2)(2(x^2 - 1) -x^2)

i expanded inside my binomial and collected like terms to x(x^2 - 1)^(-3/2) * (x^2 - 2)

and I guess since I bothered to type this out I may as well verify I would solve x(x^2 - 2) = 0 for the critical points and get the other critical points from (x^2 - 1) ^ (-3/2) = 0 due to domain restrictions.

thanks again for everything and sorry for posting something I ended up figuring out (im pretty sure anyway), but its amazing what someone verifying your right up until that point does for confidence to finish.