Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
y^2(y^2−4)=x^2(x^2−5) at point (0,−2)

Answer 1

OK SAME PROCEDURE ONLY THIS TIME EASIER SINCE WE DID IT BEFORE

Answer 2

\[y ^{2}(y ^{2}-4)=x ^{2}(x ^{2}-5)\]

Answer 3

we open up the brackets

Answer 4

do i have to use the product rule there

Answer 5

\[y ^{4}-4y ^{2}=x ^{4}-5x ^{2}\]

Answer 6

\[4y ^{3}y'-8yy'=4x ^{3}-10x\]

Answer 7

from there i plug in x and y right

Answer 8

\[y'(4y ^{3}-8y)=4x ^{3}-10x\]

Answer 9

NO PRODUCT RULE HERE

Answer 10

YOU USE PRODUCT RULE WHEN YOU HAVE XY

Answer 11

oh okay

Answer 12

LET ME GIVE YOU SOME RULES

Answer 13

\[\frac{ x }{ y }=xy ^{-1}\]

Answer 14

so even if you have x/y take it to the form above and use PRODUCT RULE IMPLICITLY

Answer 15

BY THAT I MEAN WHEN YOU DIFFERENTIATE Y DONT FORGET TO ADD Y'

Answer 16

BUT WHEN YOU DIFFERENTIATE X JUST LET IT LIKE THAT

Answer 17

ok thats helpful

Answer 18

NOW YOU SAW WHAT I DID UP THERE

JUST SUBSTITUTE YOUR VALUES X AND Y AND GET THE GRADIENT

Answer 19

THEN YOU USE GRADIENT FOUND AND THE POINTS (X,Y) AND (0,-2) AND FIND THE EQUATION OF THE LINE

Answer 20

IS IT OK

Answer 21

is 16 correct

Answer 22

LET ME CHECK

Answer 23

YOUR GRADIENT IS SUPPOSE TO BE 0

Answer 24

oh i think i see where i messed up so for my final answer i got y=-2

Answer 25

LET ME DO IT

\[y'=\frac{ 4x ^{3}-10x }{ 4y ^{3}-8y }\]

Answer 26

YOU SEE THAT X=0

Answer 27

put 0 where x is and you see that your NUMERATOR WOULD BE 0

Answer 28

IS IT FINE NOW

Answer 29

THE EQUATION OF THE TANGENT IS Y=-2

Answer 30

IS IT COOL NOW

Answer 31

yeah i see i think i put a sign wrong