MATLAB - Differential
MATLAB provides the diff command for computing symbolic derivatives. In its simplest form, you pass the function you want to differentiate to diff command as an argument.
For example, let us compute the derivative of the function f(t) = 3t2 + 2t-2
Example
Create a script file and type the following code into it −
syms t f = 3*t^2 + 2*t^(-2); diff(f)
When the above code is compiled and executed, it produces the following result −
ans = 6*t - 4/t^3
Following is Octave equivalent of the above calculation −
pkg load symbolic symbols t = sym("t"); f = 3*t^2 + 2*t^(-2); differentiate(f,t)
Octave executes the code and returns the following result −
ans = -(4.0)*t^(-3.0)+(6.0)*t
Verification of Elementary Rules of Differentiation
Let us briefly state various equations or rules for differentiation of functions and verify these rules. For this purpose, we will write f'(x) for a first order derivative and f"(x) for a second order derivative.
Following are the rules for differentiation −
Rule 1
For any functions f and g and any real numbers a and b are the derivative of the function:
h(x) = af(x) + bg(x) with respect to x is given by −
h'(x) = af'(x) + bg'(x)
Rule 2
The sum and subtraction rules state that if f and g are two functions, f' and g' are their derivatives respectively, then,
(f + g)' = f' + g'
(f - g)' = f' - g'
Rule 3
The product rule states that if f and g are two functions, f' and g' are their derivatives respectively, then,
(f.g)' = f'.g + g'.f
Rule 4
The quotient rule states that if f and g are two functions, f' and g' are their derivatives respectively, then,
(f/g)' = (f'.g - g'.f)/g2
Rule 5
The polynomial or elementary power rule states that, if y = f(x) = xn, then f' = n. x(n-1)
A direct outcome of this rule is that the derivative of any constant is zero, i.e., ify = k, any constant, then
f' = 0
Rule 6
The chain rule states that, derivative of the function of a function h(x) = f(g(x)) with respect to x is,
h'(x)= f'(g(x)).g'(x)
Example
Create a script file and type the following code into it −
syms x syms t f = (x + 2)*(x^2 + 3) der1 = diff(f) f = (t^2 + 3)*(sqrt(t) + t^3) der2 = diff(f) f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2) der3 = diff(f) f = (2*x^2 + 3*x)/(x^3 + 1) der4 = diff(f) f = (x^2 + 1)^17 der5 = diff(f) f = (t^3 + 3* t^2 + 5*t -9)^(-6) der6 = diff(f)
When you run the file, MATLAB displays the following result −
f = (x^2 + 3)*(x + 2) der1 = 2*x*(x + 2) + x^2 + 3 f = (t^(1/2) + t^3)*(t^2 + 3) der2 = (t^2 + 3)*(3*t^2 + 1/(2*t^(1/2))) + 2*t*(t^(1/2) + t^3) f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2) der3 = (2*x - 2)*(3*x^3 - 5*x^2 + 2) - (- 9*x^2 + 10*x)*(x^2 - 2*x + 1) f = (2*x^2 + 3*x)/(x^3 + 1) der4 = (4*x + 3)/(x^3 + 1) - (3*x^2*(2*x^2 + 3*x))/(x^3 + 1)^2 f = (x^2 + 1)^17 der5 = 34*x*(x^2 + 1)^16 f = 1/(t^3 + 3*t^2 + 5*t - 9)^6 der6 = -(6*(3*t^2 + 6*t + 5))/(t^3 + 3*t^2 + 5*t - 9)^7
Following is Octave equivalent of the above calculation −
pkg load symbolic symbols x=sym("x"); t=sym("t"); f = (x + 2)*(x^2 + 3) der1 = differentiate(f,x) f = (t^2 + 3)*(t^(1/2) + t^3) der2 = differentiate(f,t) f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2) der3 = differentiate(f,x) f = (2*x^2 + 3*x)/(x^3 + 1) der4 = differentiate(f,x) f = (x^2 + 1)^17 der5 = differentiate(f,x) f = (t^3 + 3* t^2 + 5*t -9)^(-6) der6 = differentiate(f,t)
Octave executes the code and returns the following result −
f = (2.0+x)*(3.0+x^(2.0)) der1 = 3.0+x^(2.0)+(2.0)*(2.0+x)*x f = (t^(3.0)+sqrt(t))*(3.0+t^(2.0)) der2 = (2.0)*(t^(3.0)+sqrt(t))*t+((3.0)*t^(2.0)+(0.5)*t^(-0.5))*(3.0+t^(2.0)) f = (1.0+x^(2.0)-(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0)) der3 = (-2.0+(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0))+((9.0)*x^(2.0)-(10.0)*x)*(1.0+x^(2.0)-(2.0)*x) f = (1.0+x^(3.0))^(-1)*((2.0)*x^(2.0)+(3.0)*x) der4 = (1.0+x^(3.0))^(-1)*(3.0+(4.0)*x)-(3.0)*(1.0+x^(3.0))^(-2)*x^(2.0)*((2.0)*x^(2.0)+(3.0)*x) f = (1.0+x^(2.0))^(17.0) der5 = (34.0)*(1.0+x^(2.0))^(16.0)*x f = (-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-6.0) der6 = -(6.0)*(-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-7.0)*(5.0+(3.0)*t^(2.0)+(6.0)*t)