Question

differentiate:
xy=[(x+3y)^4]+7

Answer 1

dy / dx or dx / dy ?

Answer 2

The total differential is: \[y \text{Dt}[x]+x \text{Dt}[y]=4 (x+3 y)^3 (\text{Dt}[x]+3 \text{Dt}[y]) \]

Answer 3

For dy/dx solve the equation for Dt[x]
\[\text{Dt}[y]=\frac{-y \text{Dt}[x]+4 (x+3 y)^3 \text{Dt}[x]}{x-12 (x+3 y)^3} \]
Factor the right side for Dt[x]
\[\text{Dt}[y]=\frac{\left(\left(4 x^3-y+36 x^2 y+108 x y^2+108 y^3\right) \text{Dt}[x]\right)}{x-12 (x+3 y)^3} \]
Divide each side by Dt[x]
\[\frac{\text{Dt}[y]}{\text{Dt}[x]}=\frac{\left(4 x^3-y+36 x^2 y+108 x y^2+108 y^3\right) }{x-12 (x+3 y)^3} \]

Answer 4

Thankyou for the medal, cara.

Answer 5

no problem thanks for ur help!! u do it a bit differently than my teacher so i was a bit confused at first but i figured it out lol thanks!

Answer 6

Your welcome.

Answer 7

As an aside,

"For dy/dx solve the equation for Dt[x]"
should have been typed:

"For dy/dx solve the equation for Dt[y]"
Very Sorry.