% Problem 7

clear
close all

% a)

% We first declare x symbolic with the syms command, and then apply diff.

syms x
diff(7*x^3 + 3*x^2 - 2*x + 1)
 
ans =
 
21*x^2+6*x-2
 
% This can easily be checked by "hand-calculation".

% b)

diff((x + 1)/(x^2 + 1))
 
ans =
 
1/(x^2+1)-2*(x+1)/(x^2+1)^2*x
 
simplify(ans)
 
ans =
 
-(x^2-1+2*x)/(x^2+1)^2
 
% Let's use the pretty command to produce an answer written in mathematical 
% notation.

pretty(ans)
 
                                   2
                                  x  - 1 + 2 x
                                - ------------
                                     2     2
                                   (x  + 1)

% c)

diff(cos(x^2 + 1))
 
ans =
 
-2*sin(x^2+1)*x
 

% d)

diff(asin(2*x + 3))
 
ans =
 
1/(-2-x^2-3*x)^(1/2)
 
pretty(ans)
 
                                      1
                              ------------------
                                     2       1/2
                              (-2 - x  - 3 x)

% e)

diff(sqrt(1 + x^4))
 
ans =
 
2/(1+x^4)^(1/2)*x^3
 
pretty(ans)
 
                                        3
                                       x
                                 2 -----------
                                         4 1/2
                                   (1 + x )

% f)

syms a
diff(a^x)
 
ans =
 
a^x*log(a)
 
% Here we have also had to declare a symbolic.

% g)

diff(atan(x))
 
ans =
 
1/(x^2+1)
 

echo off