Game Development Reference
In-Depth Information
The Power Rule
d
dt t n = nt n−1 , n is an integer.
This rule gives us the answers to the four derivatives needed above:
d
dt t 4 = 4t 3 ,
d
dt t 3 = 3t 2 ,
d
dt t 2 = 2t 1 = 2t,
d
dt t = 1t 0 = 1.
Notice in the last equation we used the identity t 0
= 1. However, even
without that identity, 24
d
dt t must be unity.
Remember that the derivative answers the question, “What is the rate of
change of the output, relative to the rate of change of the input?” In the
case of
it should be very clear that
d
dt t, the “output” and the “input” are both the variable t, and so
their rates of change are equal. Thus the ratio that defines the derivative
is equal to one.
One last comment before we plug these results into Equation (11.9) to
differentiate our polynomial. Using the identity t 0 = 1, the power rule is
brought into harmony with the constant rule:
dt k = d
d
Derivative of a constant,
using the power rule
dt (kt 0 )
Using t 0 = 1,
d
dt t 0
= k
Linear property of derivative,
−1 )]
= k[0(t
Power rule for n = 0,
= 0.
Let's get back to our fourth-degree polynomial. With the sum and
power rule at our disposal, we can make quick work of it:
x(t) = c 4 t 4 + c 3 t 3 + c 2 t 2 + c 1 t + c 0 ,
dx
dt = 4c 4 t 3 + 3c 3 t 2 + 2c 2 t + c 1 .
Below are several more examples of how the power rule can be used.
24 Be careful, t 0 is undefined when t = 0.
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