How do the derivatives of ,,, and combine with other derivative rules we have developed to expand the library of functions we can quickly differentiate?
Because each angle in standard position corresponds to one and only one point on the unit circle, the - and -coordinates of this point are each functions of . In fact, this is the very definition of and : is the -coordinate of the point on the unit circle corresponding to the angle , and is the -coordinate. From this simple definition, all of trigonometry is founded. For instance, the Fundamental Trigonometric Identity,
,
is a restatement of the Pythagorean Theorem, applied to the right triangle shown in Figure 2.4.1.
Because we know the derivatives of the sine and cosine function, we can now develop shortcut differentiation rules for the tangent, cotangent, secant, and cosecant functions. In this section’s preview activity, we work through the steps to find the derivative of .
Subsection2.4.1Derivatives of the cotangent, secant, and cosecant functions
In Preview Activity 2.4.1, we found that the derivative of the tangent function can be expressed in several ways, but most simply in terms of the secant function. Next, we develop the derivative of the cotangent function.
Notice that the derivative of the cotangent function is very similar to the derivative of the tangent function we discovered in Preview Activity 2.4.1.
Using the quotient rule we have determined the derivatives of the tangent, cotangent, secant, and cosecant functions, expanding our overall library of functions we can differentiate. Observe that just as the derivative of any polynomial function is a polynomial, and the derivative of any exponential function is another exponential function, so it is that the derivative of any basic trigonometric function is another function that consists of basic trigonometric functions. This makes sense because all trigonometric functions are periodic, and hence their derivatives will be periodic, too.
The derivative retains all of its fundamental meaning as an instantaneous rate of change and as the slope of the tangent line to the function under consideration.
When a mass hangs from a spring and is set in motion, the object’s position oscillates in a way that the size of the oscillations decrease. This is usually called a damped oscillation. Suppose that for a particular object, its displacement from equilibrium (where the object sits at rest) is modeled by the function
.
Assume that is measured in inches and in seconds. Sketch a graph of this function for to see how it represents the situation described. Then compute , state the units on this function, and explain what it tells you about the object’s motion. Finally, compute and interpret .
The derivatives of the other four trigonometric functions are
,
and.
Each derivative exists and is defined on the same domain as the original function. For example, both the tangent function and its derivative are defined for all real numbers such that , where .
The four rules for the derivatives of the tangent, cotangent, secant, and cosecant can be used along with the rules for power functions, exponential functions, and the sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to quickly differentiate a wide range of different functions.
Explain why the function that you found in (a) is almost the opposite of the sine function, but not quite. (Hint: convert all of the trigonometric functions in (a) to sines and cosines, and work to simplify. Think carefully about the domain of and the domain of .)