At this point we claim that the derivative of the sine function is another related function called the cosine function. This may seem mysterious to you since we have not defined either of these functions, but we will eventually define them.
The cosine function, whose value at argument x is generally written as cos(x) is a short way of saying the sine of the complementary angle to x. The complementary angle to x is the difference between a right angle and the angle x.
We will always try to measure angles in radians, though out of habit, old farts like myself often lapse into describing them by degrees.
Imagine we have an angle at some point P and draw a circle C around P that has radius 1. Then the size of the angle in radians is the length of the arc between the end-lines of on the circle C.
It is an important bit of folklore that the total distance around a circle of unit radius is . Thus the size of an angle is the proportion of the circle that it represents, multiplied by the factor .
We can therefore see that a straight line angle represents half a circle
so it has
angle while a right angle, which is half of a straight line, has angle . The complementary angle to is the angle .
So the derivative of the sine (written usually as sin) obeys
by the chain rule, we get
These facts about the derivatives of the sine and cosine are almost as simple as those for the exponent, and they are not difficult to use in practice. For more details about trigonometric functions, click here.
Armed with these last two facts we can use the substitution rule and our previous rule to differentiate any function we care to construct from the identity, the exponent, and the sine by arithmetic operations and substitution.
Are we done?
We are almost done. Practice using the multiple occurrence rule and the chain rule a bit and you can become an expert differentiator. But we still have to notice how to differentiate inverse functions.
7.1 Find the derivative for each of the following functions:
a. (sin x) * exp(2x)
b. x * cos 2x
c. (cos x) * sin x
7.2 Check your answers with the applet.
7.3 The derivative of the derivative of a function f is called the second
derivative of f. Find the second derivative of cos x, and also of sin 2x.
The second derivative of f is usually denoted by f ''(x) or .
7.4 What functions can you think of that obey the equation f(x) + f ''(x) = 0?
Why do you start only with the sine function exponential and identity? what happened to the cosine? Or to the other trigonometric functions? And what are those weird things on my calculator like cosh and sinh?
We don't bother with treating the cosine separately since we can define it from the sine by substitution: . The other trigonometric functions can be defined from these two. The functions with the h on the end are called hyperbolic sine and cosine. They are easily expressed in terms of the exponential function:
The other functions that appear on good calculators and are available on spreadsheets are easily constructed from those mentioned so far or from their inverses.
And what are inverses?
We will see that now, and also how to find their derivatives.