How do you implicitly differentiate csc(x^2/y^2)=e^-x-y ?

1 Answer
Mar 12, 2018

dy/dx = (y^3e^-x-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)))/(-y^3-2x^2csc((x^2)/(y^2))cot((x^2)/(y^2)))

Explanation:

d/dx (csc((x^2)/(y^2)) = e^-x -y)
First we find the derivative of the left side of the function d/dx (csc((x^2)/(y^2)) This is a composition of functions. Meaning we apply the Chain Rule. where f(x) = csc(x) and g(x) = (x^2)/(y^2) where f(g(x)) = csc((x^2)/(y^2))

d/dx f(x) = csc(x) = -csc(x)cot(x) we must substitute of x = g(x) , so d/dx csc(g(x)) = -csc((x^2)/(y^2))cot((x^2)/(y^2))

d/dx g(x) = (2xy^2-2yx^2dy/dx)/(y^4) apply the quotient rule (f'g -fg')/(g)^2 where f = x^2 and g = y^2

Furthermore, d/dx f(g(x)) = (-csc((x^2)/(y^2))cot((x^2)/(y^2)))(2xy^2-2yx^2dy/dx)/(y^4) let us simplify this function now as to make our lives easier in the following minutes.
We multiple (-csc((x^2)/(y^2))cot((x^2)/(y^2))) by both term 2xy^2 and term -2yx^2dy/dx resulting in a function similarly looking to that of

(2yx^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2))-2xy^2csc((x^2)/(y^2))cot((x^2)/(y^2)))/(y^4) we see that the top function has a like term of y , so
(2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2))-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)))/(y^3)

We are not finished we must now find the derivative of the right side of the function. d/dx e^-x - y substituting u = -x d/dx e^u = e^-x
d/dx -x = -1
hence d/dx e^u = -e^-x
d/dx -y = -dy/dx

Therefore d/dx e^-x -y = -e^-x-dy/dx

Furthermore, (2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2))-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)))/(y^3) = -e^-x-dy/dx

From here we must move dy/dx to one side of our function.
Meaning first multiple the left side of the function by y^3 moving this term to the right. resulting in,
(2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2))-2xycsc((x^2)/(y^2))cot((x^2)/(y^2))) = -y^3e^-x-y^3dy/dx

From here let us subtract 2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2)) from both sides because it has the term dy/dx resulting in,
-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)) = -y^3e^-x-y^3dy/dx-2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2))

Now let us remove any terms on the right side that aren't dy/dx , so
y^3e^-x-2xycsc((x^2)/(y^2))cot((x^2)/(y^2))= -y^3dy/dx-2x^2dy/dxcsc((x^2)/(y^2))cot((x^2)/(y^2)) From here factor out dy/dx from the right of our equation y^3e^-x-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)) = dy/dx(-y^3-2x^2csc((x^2)/(y^2))cot((x^2)/(y^2))) now divide both sides by our factored expression resulting in our final answer.
dy/dx = (y^3e^-x-2xycsc((x^2)/(y^2))cot((x^2)/(y^2)))/(-y^3-2x^2csc((x^2)/(y^2))cot((x^2)/(y^2)))