## Differentiation

This example shows how to analytically find and evaluate derivatives using Symbolic Math Toolbox™. In the example you will find the 1st and 2nd derivative of f(x) and use these derivatives to find local maxima, minima and inflection points.

### First Derivatives: Finding Local Minima and Maxima

Computing the first derivative of an expression helps you find local minima and maxima of that expression. Before creating a symbolic expression, declare symbolic variables:

syms x

By default, solutions that include imaginary components are included in the results. Here, consider only real values of x by setting the assumption that x is real:

assume(x, 'real')

As an example, create a rational expression (i.e., a fraction where the numerator and denominator are polynomial expressions).

f = (3*x^3 + 17*x^2 + 6*x + 1)/(2*x^3 - x + 3)
f =

$\frac{3 {x}^{3}+17 {x}^{2}+6 x+1}{2 {x}^{3}-x+3}$

Plotting this expression shows that the expression has horizontal and vertical asymptotes, a local minimum between -1 and 0, and a local maximum between 1 and 2:

fplot(f)
grid

To find the horizontal asymptote, compute the limits of f for x approaching positive and negative infinities. The horizontal asymptote is y = 3/2:

lim_left = limit(f, x, -inf)
lim_left =

$\frac{3}{2}$

lim_right = limit(f, x, inf)
lim_right =

$\frac{3}{2}$

Add this horizontal asymptote to the plot:

hold on
plot(xlim, [lim_right lim_right], 'LineStyle', '-.', 'Color', [0.25 0.25 0.25])

To find the vertical asymptote of f, find the poles of f:

pole_pos = poles(f, x)
pole_pos =

$-\frac{1}{6 {\left(\frac{3}{4}-\frac{\sqrt{241} \sqrt{432}}{432}\right)}^{1/3}}-{\left(\frac{3}{4}-\frac{\sqrt{241} \sqrt{432}}{432}\right)}^{1/3}$

Approximate the exact solution numerically by using the double function:

double(pole_pos)
ans = -1.2896

Now find the local minimum and maximum of f. If a point is a local extremum (either minimum or maximum), the first derivative of the expression at that point is equal to zero. Compute the derivative of f using diff:

g = diff(f, x)
g =

$\frac{9 {x}^{2}+34 x+6}{2 {x}^{3}-x+3}-\frac{\left(6 {x}^{2}-1\right) \left(3 {x}^{3}+17 {x}^{2}+6 x+1\right)}{{\left(2 {x}^{3}-x+3\right)}^{2}}$

To find the local extrema of f, solve the equation g == 0:

g0 = solve(g, x)
g0 =

Approximate the exact solution numerically by using the double function:

double(g0)
ans = 2×1

-0.1892
1.2860

The expression f has a local maximum at x = 1.286 and a local minimum at x = -0.189. Obtain the function values at these points using subs:

f0 = subs(f,x,g0)
f0 =

Approximate the exact solution numerically by using the double function on the variable f0:

double(f0)
ans = 2×1

0.1427
7.2410

Add point markers to the graph at the extrema:

plot(g0, f0, 'ok')

### Second Derivatives: Finding Inflection Points

Computing the second derivative lets you find inflection points of the expression. The most efficient way to compute second or higher-order derivatives is to use the parameter that specifies the order of the derivative:

h = diff(f, x, 2)
h =

Now Simplify that result:

h = simplify(h)
h =

$\frac{2 \left(68 {x}^{6}+90 {x}^{5}+18 {x}^{4}-699 {x}^{3}-249 {x}^{2}+63 x+172\right)}{{\left(2 {x}^{3}-x+3\right)}^{3}}$

To find inflection points of f, solve the equation h = 0. Here, use the numeric solver vpasolve to calculate floating-point approximations of the solutions:

h0 = vpasolve(h, x)
h0 =

$\left(\begin{array}{c}0.57871842655441748319601085860196\\ 1.8651543689917122385037075917613\\ -1.4228127856020972275345064554049-1.8180342567480118987898749770461 \mathrm{i}\\ -1.4228127856020972275345064554049+1.8180342567480118987898749770461 \mathrm{i}\\ -0.46088831805332057449182335801198+0.47672261854520359440077796751805 \mathrm{i}\\ -0.46088831805332057449182335801198-0.47672261854520359440077796751805 \mathrm{i}\end{array}\right)$

The expression f has two inflection points: x = 1.865 and x = 0.579. Note that vpasolve also returns complex solutions. Discard those:

h0(imag(h0)~=0) = []
h0 =

$\left(\begin{array}{c}0.57871842655441748319601085860196\\ 1.8651543689917122385037075917613\end{array}\right)$

Add markers to the plot showing the inflection points:

plot(h0, subs(f,x,h0), '*k')
hold off

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