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14.6 Some Derivative Calculations

14.46   Example (Derivative of .) We know that

If we differentiate both sides of this equation, we get

i.e.

14.47   Example (Derivative of ) Let be any real number and let for all . Then

so by the chain rule

(Here I have used the result of exercise 14.37.) Thus the formula

which we have known for quite a while for rational exponents, is actually valid for all real exponents.

14.48   Exercise (Derivative of .) Let . Show that

for all

14.49   Example (Derivative of .)

Hence

14.50   Example (Derivative of .) Let be defined by

We have

so has an inverse function which is denoted by . By the inverse function theorem is differentiable on . and we have

By the chain rule

Now since the sine function is positive on we get

for all , so

Thus

14.51   Exercise (Derivative of .) Let

Show that has an inverse function that is differentiable on the interior of its domain. This inverse functions is called . Describe the domain of , sketch the graphs of and of , and show that

14.52   Example (Derivative of .) Let

Then T is continuous, and the image of is unbounded both above and below, so image() = R. Also

so has an inverse function, which we denote by .
For all

so by the chain rule

Now

so

Thus

14.53   Exercise (Derivative of arccot.) Let

Show that has an inverse function arccot, and that

What is ? Sketch the graphs of and of arccot.

Remark The first person to give a name to the inverse trigonometric functions was Daniel Bernoulli (1700-1792) who used for in 1729. Other early notations included arc(cos. = ) and ang(cos. = )[15, page 175]. Many calculators and some calculus books use to denote arccos. (If you use your calculator to find inverse trigonometric functions, make sure that you set the degree-radian-grad mode to radians.)

14.54   Exercise. A Calculate the derivatives of the following functions, and simplify your answers (Here is a constant.)
a)
.
b)
.
c)
.
d)
.
e)
.
f)
.
g)
.

14.55   Exercise. A Let

Calculate the derivative of . What is the domain of this function? Sketch the graph of .

14.56   Exercise (Hyperbolic functions.) We define functions and on R by

These functions are called the hyperbolic sine and the hyperbolic cosine respectively. Show that

and

Calculate

and simplify your answer as much as you can. What conclusion can you draw from your answer? Sketch the graphs of and on one set of coordinate axes.

Next: 15. The Second Derivative Up: 14. The Inverse Function Previous: 14.5 Inverse Function Theorems   Index
Ray Mayer 2007-09-07