Game Development Reference
In-Depth Information
Derivatives of Sine and Cosine
d
dx
sinx = cosx,
d
dx
cosx = − sinx.
The derivatives of the sine and cosine functions will become useful in later
sections.
Now let's look at one more important special function that will play
an important role later in this topic, which will be convenient to be able
to differentiate, and which also happens to have a nice, tidy Taylor series.
The function we're referring to is the exponential function, denoted e
x
. The
mathematical constant e ≈ 2.718282 has many well known and interesting
properties, and pops up in all sorts of problems from finance to signal
processing. Much of e's special status is related to the unique nature of the
function e
x
. One manifestation of this unique nature is that e
x
has such a
beautiful Taylor series:
Taylor series of e
X
e
x
= 1 + x +
x
2
2!
+
x
3
3!
+
x
4
4!
+
x
5
5!
+
(11.12)
Taking the derivative gives us
1 + x +
x
2
2!
+
x
3
3!
+
x
4
4!
+
x
5
dx
e
x
=
d
d
5!
+
dx
= 0 + 1 +
x
1!
+
x
2
2!
+
x
3
3!
+
x
4
4!
+
= 1 + x +
x
2
2!
+
x
3
3!
+
x
4
4!
+
But this result is equivalent to the definition of e
x
in Equation (11.12); the
only difference between them is the cosmetic issue of when to stop listing
terms explicitly and end with the “ ”. In other words, the exponential
function is its own derivative: d/dx e
x
= e
x
. The exponential function is
the only function that can boast this unique property. (To be more precise,
any multiple of the exponential function, including zero, has this quality.)
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