Question

Find the derivative y' by implicit differentiation if xy=sin(pi*y). Find the slope of the tangent line to xy=sin(pi*y) at point (x,y)=(0,1)

Answer 1

did you find derivative of both sides yet?

Answer 2

xy is a product so use the product rule to find (xy)'

Answer 3

so far i got that y'=cos(pi*y), but im not sure if thats right

Answer 4

I'm not sure how you got that answer.
Anyways I'm asking you to find (xy)'
use the product rule since you have a product

Answer 5

would it be y+xy'

Answer 6

right (xy)'=y+xy'

Answer 7

now we need to find (sin(pi*y))'

Answer 8

you need the chain rule and constant multiply rule here

Answer 9

what is (pi *y)' ?

Answer 10

just y+pi*y'?

Answer 11

looks like you tried to use the product rule but did so incorrectly

Answer 12

wait no y

Answer 13

opps um cos(pi*y)*y'

Answer 14

You don't need the product rule to find derivative of pi*y

like you have a constant times a function

derivative will be just that constant times the derivative of that function
this is what the constant multiple rule says


also we could do the product rule but watch what happens
\[\frac{d(\pi y)}{dx}=\pi \frac{dy}{dx}+y \frac{d \pi}{dx}=\pi \frac{dy}{dx}+y(0)=\pi \frac{dy}{dx}=\pi y'\]

Answer 15

\[\frac{d(\sin(\pi y))}{dx}=\pi y' \cos(\pi y) \]

Answer 16

so you have both sides after the differentiating step

Answer 17

now you need to solve for y'

Answer 18

quick question, pi isnt a constant?

Answer 19

pi is totally a constant

Answer 20

pi is always 3.14159....

Answer 21

it never changes value

Answer 22

that is why it is called a constant
because it constantly remains the same value forever and ever

Answer 23

so when you use the chain rule, since pi is a constant how come pi is still included in the answer. Because isn't the derivative of a constant 0?

Answer 24

did you see what happened when i used the product rule?
also do you not know the constant multiple rule?

Answer 25

I copy and pasted what I said above
\[\frac{d(\pi y)}{dx}=\pi \frac{dy}{dx}+y \frac{d \pi}{dx}=\pi \frac{dy}{dx}+y(0)=\pi \frac{dy}{dx}=\pi y'\]
The only reason I used the product rule here is because you tried to use it on this and failed

Answer 26

you could have said well pi is a constant
and since it is being multiplied to a function I could use the constant multiple rule
which says
(cy)'=cy'
where c is a constant

Answer 27

\[(\pi y)'=\pi y'\]

Answer 28

also you the derivative of a constant is zero that is why i put dpi/dx=0 when using the product rule