11DIFFERENTIATION - Department of Mathematics
11DIFFERENTIATION - Department of Mathematics
11DIFFERENTIATION - Department of Mathematics
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760 11 DIFFERENTIATION<br />
EXAMPLE 1<br />
SOLUTION ✔<br />
Finding dy<br />
by Implicit<br />
dx<br />
Differentiation<br />
EXAMPLE 2<br />
SOLUTION ✔<br />
Find dy<br />
dx given the equation y2 x.<br />
Differentiating both sides <strong>of</strong> the equation with respect to x, we obtain<br />
d<br />
dx (y2 ) d<br />
dx (x)<br />
To carry out the differentiation <strong>of</strong> the term d<br />
dx y2 , we note that y is a function<br />
<strong>of</strong> x. Writing y f(x) to remind us <strong>of</strong> this fact, we find that<br />
d<br />
dx (y2 ) d<br />
[ f(x)]2 [Writing y f(x)]<br />
dx<br />
2f(x)f(x) (Using the chain rule)<br />
2y dy<br />
dx<br />
Therefore, the equation<br />
is equivalent to<br />
[Returning to using y instead <strong>of</strong> f(x)]<br />
d<br />
dx (y2 ) d<br />
dx (x)<br />
2y dy<br />
1<br />
dx<br />
Solving for dy<br />
dx yields<br />
dy 1<br />
<br />
dx 2y <br />
Before considering other examples, let us summarize the important steps<br />
involved in implicit differentiation. (Here we assume that dy/dx exists.)<br />
1. Differentiate both sides <strong>of</strong> the equation with respect to x. (Make sure that<br />
the derivative <strong>of</strong> any term involving y includes the factor dy/dx.)<br />
2. Solve the resulting equation for dy/dx in terms <strong>of</strong> x and y.<br />
Find dy/dx given the equation<br />
y3 y 2x3 x 8<br />
Differentiating both sides <strong>of</strong> the given equation with respect to x, we obtain<br />
d<br />
dx (y3 y 2x3 x) d<br />
dx (8)<br />
d<br />
dx (y3 ) d d<br />
(y) <br />
dx dx (2x3 ) d<br />
(x) 0<br />
dx
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