Question

help please: Use implicit differentiation to find the equation of the tangent line to the curve at the point. (attached below)

Answer 1

i know i have to use implicit differentiation, then plug into derivative to find slope, plug slope and given points to point-slope formula.

Answer 2

Problem I have is step 1

Answer 3

take the derivative with respect to x. use the product rule
for example, on the xy term you do this
\[ \frac{d}{dx}xy = x \frac{d}{dx}y+ y \frac{d}{dx}x \\
= x \frac{dy}{dx}+ y \frac{dx}{dx} \\ = x \frac{dy}{dx}+ y\]

use the same idea on \(xy^3\)

Answer 4

i got x(3y*dy/dx) +y^3

Answer 5

@phi

Answer 6

so.. 3xy*dydx +y^3

Answer 7

dy/dx**

Answer 8

you mean 3 x y^2 dy/dx for the first part

d/dx of y^3 is 3 y^2 dy/dx

Answer 9

okay im lost with the second part.

Answer 10

i know that you took the derivative of xy^3= 3xy^2

Answer 11

+xy= 1?

Answer 12

*dy/dx

Answer 13

from the beginning
\[ \frac{d}{dx}\left(xy^3 + xy = 4\right) \\ x \frac{d}{dx}y^3 + y^3 \frac{d}{dx}x+ x\frac{d}{dx}y+ y \frac{d}{dx}x=\frac{d}{dx}4\]

that simplifies to
\[ x \frac{d}{dx}y^3 + y^3+ x\frac{dy}{dx}+ y =0\]

now you can do the d/dx of y^3 , right ?

Answer 14

yes.

Answer 15

use the power rule
\[ \frac{d y^n}{dx} = n y^{n-1} \frac{dy}{dx} \]

Answer 16

this isn't applying chain rule or is it? because i understand what youre doing here makes sense/

Answer 17

yes, it is the chain rule

Answer 18

normally you do d/dx of x^3 and you get 3x^2 dx/dx
but the dx/dx is left off and the rule looks like
\[ \frac{d\ x^3}{dx} = 3 x^2 \]

Answer 19

okay, let me give this a try.

Answer 20

okay I got this down: x*3y^2(dy/dx) +3y +x(dy/dx)+y=0

Answer 21

you mean y^3 not 3y , right ?

Answer 22

oh so only find the derivative of [y^3]...

Answer 23

thanks btw! :)

Answer 24

then i should just plot the points for x and y

Answer 25

wait a min. what should i do with dy/dx?

Answer 26

solve for dy/dx and then sub in for x and y to get the value of dy/dx at the point (2,1)

Answer 27

how?

Answer 28

use algebra

Answer 29

question: is it possible to divide this: -4x/-2y ????

Answer 30

how the hell can someone divide this and get 2x/y?????

Answer 31

this is so frustrating, please help me understand this.

Answer 32

@asnaseer

Answer 33

what part is confusing ?

you have
\[3 x y^2\frac{dy}{dx} + y^3+ x\frac{dy}{dx}+ y =0 \]
plug in x=2 and y =1 and solve for dy/dx (which is the slope)

Answer 34

at the point (2,1)

Answer 35

no no, i know that. i was just looking at an example and i see that they divide -4x/-2y and got 2x/y

Answer 36

it frustrated me to see this and not know how they even possibly do it.

Answer 37

are you asking how to get from
\[ \frac{-4x}{-2y} = \frac{2x}{y} \]
?
-4/-2 is +2

Answer 38

yes but with diferet variables?? 0.o

Answer 39

different*

Answer 40

what is the question ?

Answer 41

oh the slope is -1/4! =]

Answer 42

yes. also, the tangent line goes through point (2,1)
so you can find the intercept.

Answer 43

cool, how do you know this?

Answer 44

see
http://www.khanacademy.org/math/calculus/differential-calculus/implicit_differentiation/v/implicit-differentiation-1
and the following videos , especially
http://www.khanacademy.org/math/calculus/differential-calculus/implicit_differentiation/v/finding-slope-of-tangent-line-with-implicit-differentiation

Answer 45

okie doke

Answer 46

thanks @phi you helped me so much! ^_^ so thankful

Answer 47

they asked you to find the tangent line through point (2,1)
you found its slope = -1/4
you know it goes through 2,1
so in point-slope form you have
y- 1 = (-1/4)(x-2)
which you can change to slope-intercept form

Answer 48

y=-1/4x+3/2

Answer 49

yes