Question

How to find the derivative of y = sqr 3x/x^-4 at x=3 using the chain rule.

Answer 1

\[\frac{\sqrt{x}}{x^{-4}}\]?

Answer 2

Is that:\[y=\frac{ \sqrt{3x} }{ x^{-4} }\]

Answer 3

the bottom is x^2-4

Answer 4

i did it but i'm stuck at one step

Answer 5

So it's:\[y=\frac{ \sqrt{3x} }{ x^2 -4 }\]

?

Answer 6

yes

Answer 7

so far I have y prime = (3/2 x^-1/2)(x^2-4)-(2x)(3x^1/2)/ (x^2-4)^2

Went i sub in x=3 into (3/2 x^-1/2) I don't know how to solve it so the whole equation works out to -31/50

Answer 8

We use quotient rule here as well as chain rule.\[y'=\frac{( \frac{ 1 }{ 2 }(3x)^{-\frac{ 1 }{ 2 }}*3)(x^2-4)-(\sqrt{3x})(2x) }{ (x^2-4)^2 }\]\[=\frac{ \frac{ 3x^2-12 }{ 2\sqrt{3x} }- 2x \sqrt {3x}}{ (x^2-4)^2 }\]\[=\frac{ \frac{ 3x^2-12 }{ 2\sqrt{3x} }- \frac{ 12x^2 }{ 2\sqrt{3x} }}{ (x^2-4)^2 }\]\[=\frac{ \frac{ -9x^2-12 }{ 2\sqrt{3x} }}{ (x^2-4)^2 }\]\[={ \frac{ -9x^2-12 }{ 2\sqrt{3x}(x^2-4)^2 }} \rightarrow y'(3)={ \frac{ -9(3)^2-12 }{ 2\sqrt{3(3)}((3)^2-4)^2 }}={ \frac{ -81-12 }{6(25) }}=\frac{ -93 }{ 150 }=-\frac{ 31 }{ 50 }\] @0213

Answer 9

@satellite73

Answer 10

in the first bracket were did you get the second 3?

Answer 11

Chain rule, we first differentiate sqrt(3x) using power-rule, but then we also have to multipy it by the derivative of whatever is inside the square root, this is where the chain rules comes. If it was just an x inside the square root, nothing would change because derivative of x is just 1. But the derivative of 3x is 3 so after differentiating sqrt(3x), we have to multiply that by the derivative of whatever is inside the square root which is 3.

Answer 12

@0213
Get it?

Answer 13

so if i had 4x-1 inside the sqr, then i would multiply by 4. I would have to use the power rule, then the chain rule.

Answer 14

Exactly.

Answer 15

oh ok. when you multiply the first two brackets together, do you get a common denominator?

Answer 16

In the first step when I multiply the first 2 brackets, the 3 goes to the numerator, root of 3x goes to the denominator. Then multiply the top with x^2-4. Now the second bracket after the minus sign needs to have the same denominator so I can subtract the two, so I multiplied its top and bottom by 2sqrt(3x) then subtracted the fractions.

Answer 17

so you multiply the top with x^2-4, then find a common denominator for the second set of brackets and then solve

Answer 18

Yup. @0213

Answer 19

It's just quotient rule and chain rule. That's it.

Answer 20

when i was doing it, i didn't know that i have to take the derivative of 3x, so i was not including the 3.

Answer 21

Btw 0213, I would appreciate a medal lol. @0213

Answer 22

derivative of sqrt(f) = f ' / 2 sqrt(f)

Answer 23

But boy, that was a lot of typing I tell you =.=

Answer 24

lol, sorry. thank you so much though

Answer 25

it really helped. I just hate using the quotient rule since i always forget to do something and mistakes come up.