Derivatives 215 problem 01.mws

1. Differentiating Inverse trig functions.

Maple allows one to differentiate functions which are given by expressions and functions which are defined by names.   To differentiate a function which is given by an expression, use the diff command. For example,

> diff(3*x^2+2*x-1,x);

6*x+2

computes the derivative of the function determined by the expression 3*x^2+2*x-1 . Note that you must tell Maple what the variable of the function is in order to differentiate expressions. To differentiate a function which you have defined in Maple, you proceed as follows.

> f:=x->3*x^2+2*x-1;

f := proc (x) options operator, arrow; 3*x^2+2*x-1 ...

> D(f);

proc (x) options operator, arrow; 6*x+2 end proc

> D(f)(x);

6*x+2

Notice that if you just type in D(f) , Maple tells you the rule for computing the derivative as a function, while if you evaluate D(f)(x) , it tells you the expression for the derivative at the value x . Of course, since the variable name is not part of the definition of the function, you could also evaluate it at any value, as illustrated below..

> D(f)(t);

6*t+2

Let us differentiate an inverse trig function.

> D(arcsin)(x);

1/(sqrt(1-x^2))

Submission:

(a) Differentiate the arcsec function in Maple. Explain how the formula you obtain is equivalent to the formula in the book.

(b) Define the function f(x) = arcsin(x)+arccos(x) in Maple, and compute its derivative.

(c) What can we conclude about the function from the mean value theorem?

(d) What is the domain of the function?

(e) Compute the value of the function at x = 0 , and explain why this computation tells you the value of the function at any point in its domain.

(f) Plot this function over its domain.

Submission worksheet: