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deryn=dyn./dxn;
plot(n,deryn)

The limit of the deryn’s sequence is clearly equal to 1, the value of this func-
tion derivative at 0.
NOTE The choice of N should always be such that dxn is larger than the
53
machine precision; that is, N < 53, since (1/2) ≈ 10 .
–16

In-Class Exercises
Find numerically, to one part per 10,000 accuracy, the derivatives of the fol-
lowing functions at the indicated points:

3
4
Pb. 4.6 x (cos ( x) sin(− 2 x)) at x → π
exp(x + ) 3
2
Pb. 4.7 at x → 0
(2 + cos ( ))x
2
(1+ sin ( ))x
2
Pb. 4.8 at x → π / 2
(2 − cos ( ))x
3
 x − 12
/ 
Pb. 4.9 ln  x + 1  at x → 1
−1
Pb. 4.10 tan (x 2 + ) 3 at → 0x

Example 4.3
2
Plot the derivative of the function x sin(x) over the interval 0 ≤ x ≤ 2π.
Solution: Edit and execute the following script M-file:

dx=10^(-4);
x=0:dx:2*pi+dx;
df=diff(sin(x).*x.^2)/dx;
plot(0:dx:2+pi,df)

where diff is a MATLAB command, which when acting on an array X, gives
the new array [X(2) – X(1)X(3) – X(2) … X(n) – X(n – 1)], whose length is one
unit shorter than the array X.
The accuracy of the above algorithm depends on the choice of dx. Ideally,
the smaller it is, the more accurate the result. However, using any computer,
we should always choose a dx that is larger than the machine precision, while

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