11DIFFERENTIATION - Department of Mathematics

Continuing, we find
3. N(t) 6t 2 6t 4
N(t) 12t 6 6(2t 1)
11.6 Implicit Differentiation and Related Rates
11.6 IMPLICIT DIFFERENTIATION AND RELATED RATES 759
f (x) (2)(1 x) 3
2(1 x) 3
2
(1 x) 3
f (x) 2(3)(1 x) 4
6(1 x) 4
6
(1 x) 4
Therefore, N(2) 30 and N(8) 102. The results of our computations reveal
that at the end of year 2, the rate of growth of the turtle population is increasing
at the rate of 30 turtles/year/year. At the end of year 8, the rate is increasing at
the rate of 102 turtles/year/year. Clearly, the conservation measures are paying
off handsomely.
D IFFERENTIATING I MPLICITLY
Up to now we have dealt with functions expressed in the form y f(x); that
is, the dependent variable y is expressed explicitly in terms of the independent
variable x. However, not all functions are expressed in this form. Consider,
for example, the equation
x2y y x2 1 0 (8)
This equation does express y implicitly as a function of x. In fact, solving (8)
for y in terms of x, we obtain
(x2 1)y x2 1 (Implicit equation)
y f(x) x2 1
x 2 1
(Explicit equation)
which gives an explicit representation of f.
Next, consider the equation
y4 y3 y 2x3 x 8
When certain restrictions are placed on x and y, this equation defines y as a
function of x. But in this instance, we would be hard pressed to find y explicitly
in terms of x. The following question arises naturally: How does one go about
computing dy/dx in this case?
As it turns out, thanks to the chain rule, a method does exist for computing
the derivative of a function directly from the implicit equation defining the
function. This method is called implicit differentiation and is demonstrated
in the next several examples.