Mathematics- Implicit Differentiation

Question

Find dy/dx for
a)e^cosx + e^siny = 1/4
b) e^2x+3y = x^2 -ln (xy^3)

Find d^2/dx^2 for:
x^3+y^3=8

Answer 1

a)  using implicit differentiation on the given equation, we get

e^{cos x}(-sin x) + e^{sin y}(cos y)(y') = 0 and now just solve for y' (a.k.a. dy/dx)

 

Other equations can be treated similarly.