Question

Second derivative!

Answer 1

\[sinx-4cosy=y\]

Answer 2

Determine \[d^2y/dx^2\]

Answer 3

-take derivative
- solve for dy/dx
-take derivative again
-substitute in for dy/dx to get 2nd derivative in terms of x and y

Answer 4

the first derivative i got
cosx/ (1-4siny)

Answer 5

correct

Answer 6

yeah but when trying to get the next derivative, i get mixed up and lost

Answer 7

ok did you use the quotient rule?

Answer 8

yeah

Answer 9

for that I got the equation :
\[\frac{ (-sinx)(1-4siny)-(-4cosy)(cosx) }{ (1-4siny)^2 }\]

Answer 10

\[y'' = \frac{(-\sin x)(1-4\sin y) - \cos x (-4\cos y)y'}{(1-4\sin y)^{2}}\]

Answer 11

oh right right

Answer 12

so could you go any further? or leave it like that?

Answer 13

no now you must sub in 1st derivative (y') into 2nd derivative function

Answer 14

\[\frac{4\sin x \sin y -\sin x+4\cos x \cos y(\frac{\cos x}{1-4\sin y})}{(1-4\sin y)^{2}}\]

Answer 15

then from there what should i do, foil?

Answer 16

or get rid of the fraction in the numerator

Answer 17

yeah now its a lot of simplifying so answer is 1 single fraction

Answer 18

to do get rid of it, should i just flip the fraction

Answer 19

hmm flip the fraction..no i would combine numerator into single fraction then ....
\[\frac{\frac{a}{b}}{c} = \frac{a}{bc}\]

Answer 20

is it -sinx/ 1-4siny

Answer 21

i dont follow? sorry i should have left it factored, anyway let me show you
\[(-\sin x)(1-4\sin y) +4\cos x \cos y(\frac{\cos x}{1-4\sin y}) = \frac{(-\sin x)(1-4\sin y)^{2} +4\cos^{2} x \cos y}{1-4\sin y}\]

Answer 22

sorryi cant see the end of that equation

Answer 23

its cos y

Answer 24

\[\frac{ (-sinx)(1-4siny)+4cosxcosy \times cosx) }{ (1-4siny)^3}\]

Answer 25

is what i got

Answer 26

yep the (1-4sin) is squared on top and its more simplified if you write cos*cos as cos^2

Answer 27

how is it squared on top

Answer 28

because it was multiplied by the original (1-4sin) to get common denominator of (1-4sin)
same reasoning behind.... x + 1/x = (x^2 +1)/x

Answer 29

so i get
\[\frac{ (-sinx)(1-4siny)^2+(4\cos^2x)(cosy) }{ (1-4siny)^3 }\]

Answer 30

yes

Answer 31

do i simplify further

Answer 32

you can try expanding and see if anything cancels but i would just leave it as is

Answer 33

i dont think you can, i'll leave it as is then x3