Implicit Differentiation Example
dy
How would we find y � = dx
if y 4 + xy 2 − 2 = 0?
We could use a trick to solve this explicitly — think of the above equation
as a quadratic equation in the variable y 2 then apply the quadratic formula:
√
−x ± x2 + 8
2
y
=
,
2
so
�
√
−x ± x2 + 8
y = ±
.
2
Since we see ± twice in this equation, there are four possible branches to
consider. This means that to be thorough we’d want to compute four different
derivatives. This is a lot of work.
dy
Instead, we can compute dx
using implicit differentiation. As always, we
d
start by applying dx to both sides:
d 4
(y + xy 2 − 2)
dx
d 4
d
d
(y ) +
(xy 2 ) −
2
dx
dx
dx
dy
dy
4y 3
+ (y 2 + x · 2y ) − 0
dx
dx
dy
dy
4y 3
+ 2xy
dx
dx
dy
(4y 3 + 2xy)
dx
dy
dx
use
=
d
0
dx
=
0
=
0
=
−y 2
=
−y 2
=
−y 2
+ 2xy
4y 3
In lecture Professor Jerison used the shorthand y � for the derivative; here we
dy
dx to make it clear that we are differentiating with respect to x.
1
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18.01SC Single Variable Calculus��
Fall 2010 ��
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