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Contents
Implicit Differentiation
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Implicit Differentiation 
Implicit
Function
An
implicit function is one that cannot be easily manipulated to the form y = f(x).
An
example is x2 + y2 + xy = 4
Implicit
Differentiation
To
find dy/dx, every term in the equation is differentiated with respect to
x. This is
easy for the terms in x, but what about the terms in y, since the
differentiation is with respect to x? This
is where the chain rule comes in. The following result is used:
Consider
the above equation x2 + y2 + xy = 4. We shall use this
to illustrate the idea of differentiation of implicit functions. Differentiate
each term:
d/dx (x2) = 2x
d/dx (y2) = 2y (dy/dx)
use the
chain rule as stated above
d/dx (xy) = y + x (dy/dx) chain
and product rules
d/dx (4) = 0
Hence,
the differentiated equation is
2x + 2y (dy/dx) + y + x (dy/dx) = 0
Rearrange
such that dy/dx in on the left-hand side:
dy/dx (x + 2y) = -(2x + y)
Note
that the derivative is in terms of both x and y.
 Find
the derivatives of:
(i) x2 + y3 + 2xy2 = y + 4
(ii) 3y - 4x3 + 2xy = 0
Solution:
(i)
Differentiating w.r.t x,
(ii)
Differentiating w.r.t x,
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