Question

find the derivative of the function f(x)= (x + 5x^1/2) / (5x - 4)

work that I've done so far:

quotient theorem -
[f(x)/g(x)] = (g(x)f`(x) - f(x)g`(x)) / [g(x)]^2

so:

(5x-4)(1+5^(1/2)(1/2)x^(-1/2)) - (x + 5x^(1/2))(5) / (5x-4)^2

I've tried distributing this out but got a really sloppy answer and I can't help but feel I did something wrong?

Answer 1

Regarding "quotient theorem -
[f(x)/g(x)] = (g(x)f`(x) - f(x)g`(x)) / [g(x)]^2" ... Please be certain to use additional parentheses so that there's no doubt that you're dividing everything by [g(x)]^2.

Now for a look at your actual post:

Answer 2

\[ f(x)= (x + 5x^1/2) / (5x - 4)\]

Answer 3

I typed this into Equation Editor and see that E. E. misterprets your exponent of (1/2). So I'll correct that here:\[ f(x)= (x + 5x^1/2) / (5x - 4)\rightarrow f(x)= (x + 5x ^{1/2}) / (5x - 4)\]

Answer 4

Now, let's write f and g and their derivatives:\[u(x)=x+5x ^{1/2};~ u'(x)=1+(1/2)x ^{-1/2}\]

Answer 5

\[v(x)=5x-4;~v'(x)=5\]

Answer 6

Are you OK with this so far? If so, compare my results with yours for verification.

Answer 7

shouldn't u`(x) be: 1+ 5^(1/2) (1/2)x^(-1/2)?

Answer 8

I'm not sure if I typed the problem correctly. f(x) should be x + squareroot of 5x)

Answer 9

\[x+\sqrt{5x}\] deriving this becomes \[1 + \sqrt{5} (1/2)x ^{-1/2}\] right?

Answer 10

because I can rewrite square root of 5x as \[\sqrt{5} \times \sqrt{x} \] and derive from there?

Answer 11

yes

Answer 12

Forgive me for correcting your description, but the proper word for "finding the derivative" is "differentiate." "derive" has a different meaning in mathematics.

Answer 13

woops sorry about that

Answer 14

I typed out u, v, u' and v' for you. Make corrections as necessary and retype u, v, u' and v' back to me and phi. Then we can move forward. OK?

Answer 15

you could also do this
\[\frac{d}{dx} \left(5x\right)^{\frac{1}{2}}=\frac{1}{2} \left(5x\right)^{-\frac{1}{2}}\frac{d}{dx}5x \\
= \frac{1}{2} \left(5x\right)^{-\frac{1}{2}} \cdot 5
\]

which can be written as
\[ \frac{5}{2\sqrt{5x} } = \frac{\sqrt{5}}{2\sqrt{x} } \]

Answer 16

u(x) = \[x + \sqrt{5x}\]
u`(x) = \[1 + \sqrt{5}(1/2)x ^{-1/2}\]
v(x) = 5x - 4

v`(x)= 5

Answer 17

yes

Answer 18

\[((5x-4)(1+\sqrt{5}(1/2)x ^{-1/2}) - (x + \sqrt{5x})(5))/ (5x-4)^{2}\]
so plugging this into the quotient rule we get this?

Answer 19

yes. it might simplify a little

Answer 20

Yes, and in addition to phi's suggestion, I'd like to emphasize again how important it is that you use parentheses to indicate exactly what you're dividing by \[(5x-4)^2.\] Can't stress this enough.

Answer 21

(You have already done this, so interpret my last comment as positive reinforcement!)

Answer 22

at this point I would try distributing

Answer 23

but how would you distribute 5x to (1+/sqrt{5} (1/2)x^(-1/2)?

Answer 24

\[ (5x−4)\left( 1 +\frac{\sqrt{5}}{2\sqrt{x}} \right) = 5x + \frac{5\sqrt{5x}}{2} -4 -\frac{2\sqrt{5}}{\sqrt{x}} \]

for the first term on the top.
the second term gives
\[ -5x - 5 \sqrt{5x} \]

the 5x - 5x cancels.
we can do a little more, but it is not going to look nice, no matter what.