Find dy/dx for the folowing

Question

Find dy/dx for the following:

a. xy=cot (xy)

b. x^2y=y^9x

Answer 1

Hello J.M.,

For your question:

a. This one will definitely require implicit differentiation as 'y' is present on both the left hand side and right hand side of the expression.

So,

d/dx(xy) = d/dx(cot(xy))

I will move forward with the right hand side first and leave the left hand side for your exercise.

d/dx(cot(xy)) = -csc^2(xy)*d/dx(xy) by chain rule since 'xy' is contained inside of cot()

= -csc^2(xy)*(d/dx(x)*y + d/dx(y)*x) since 'x' and 'y' are two independent functions of 'x'

= -csc^2(xy)*(y + d/dx(y)*x) simplified a little bit. Now, the dy/dx is present and can be isolated.

b. I'm going to assume you meant: (x^2)*y = (y^9)*x

So, I will move forward with the right hand side first and leave the left hand side for you.

d/dx((y^9)*x) = d/dx(y^9)*x + d/dx(x)*y^9 since 'y^9' and 'x' are independent functions of 'x'

= (9y^8)*(dy/dx)*x + (y^9) chain rule with implicit differentiation for d/dx(y^9)

Now a dy/dx is present and can be isolated.

Hope this helps!