E-StoreShortcut for Differentiation of Implicit functions
Any function which have x and y as variables and is not immediately solvable for y is called Implicit functions.
They are given by f(x,y)=0.
d {f(x,y)}/dx
= -(df/dx)/(df/dy)
where df/dx is partial differentiation of given function ' f ' w.r.t. x (i.e. differentiate f w.r.t. x keeping y constant) and df/dy is partial differentiation of given funtion w.r.t y ( i.e. differentiate f w.r.t. y keeping x constant).
Illustration : If x2 + y2 + xy =2 find dy/dx.
Sol: f(x,y)= x2 + y2 +xy -2= 0
dy/dx = -(df/dx)/(df/dy) ...................(1)
df/dx= 2x + 0 + y - 0 = 2x + y
df/dy=2y + x
Substituting in (1),
dy/dx = - (2x +y) / ( 2y + x)
Many long calculations can be done easily by this method. Hope this method helps you.
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