【Advanced Chapter】Detailed Explanation of MATLAB Symbolic Calculus Calculation Functions

# [Advanced] Detailed Explanation of MATLAB Symbolic Calculus Functions

## 2.1 Symbolic Differentiation Function

### 2.1.1 Using the diff() Function

The `diff()` function is used to calculate the derivative of a symbolic expression. Its syntax is as follows:

diff(expr, var)

Where:

* `expr`: The symbolic expression whose derivative is to be calculated.
* `var`: The variable with respect to which the derivative is taken.

For example, calculating the derivative of the expression `x^2 + 2x` with respect to `x`:

>> syms x;
>> expr = x^2 + 2*x;
>> d = diff(expr, x);
>> d =
     2*x + 2

## 2. Symbolic Calculus Function Fundamentals

### 2.1 Symbolic Differentiation Function

#### 2.1.1 Using the diff() Function

MATLAB provides the `diff()` function to calculate the derivative of a symbolic expression. Its syntax is as follows:

diff(expr, var)

Where:

* `expr`: The symbolic expression to be differentiated.
* `var`: The variable with respect to which the differentiation is performed.

For example, calculating the derivative of `x^2` with respect to `x`:

syms x;
y = x^2;
dy_dx = diff(y, x);
disp(dy_dx);

Output:

2*x

#### 2.1.2 Differentiation Rules and Techniques

MATLAB supports various differentiation rules, including:

***Power Rule:** `d/dx(x^n) = n*x^(n-1)`
***Product Rule:** `d/dx(u*v) = u*dv/dx + v*du/dx`
***Quotient Rule:** `d/dx(u/v) = (v*du/dx - u*dv/dx) / v^2`
***Chain Rule:** `d/dx(f(g(x))) = f'(g(x)) * g'(x)`

These rules can be automatically applied using the `diff()` function. For example, calculating the derivative of `sin(x^2)` with respect to `x`:

syms x;
y = sin(x^2);
dy_dx = diff(y, x);
disp(dy_dx);

Output:

2*x*cos(x^2)

### 2.2 Symbolic Integration Function

#### 2.2.1 Using the int() Function

MATLAB provides the `int()` function for calculating the integral of a symbolic expression. Its syntax is as follows:

int(expr, var)

Where:

* `expr`: The symbolic expression to be integrated.
* `var`: The variable to integrate over.

For example, calculating the integral of `x^2` with respect to `x`:

syms x;
y = x^2;
int_y_dx = int(y, x);
disp(int_y_dx);

Output:

x^3/3 + C

Where `C` is the constant of integration.

#### 2.2.2 Integration Rules and Techniques

MATLAB supports various integration rules, including:

***Power Rule:** `∫x^n dx = x^(n+1)/(n+1)`
***Product Rule:** `∫u*v dx = u*∫v dx - ∫(∫v du) dx`
***Integration by Parts:** `∫u dv = uv - ∫v du`
***Substitution:** `∫f(g(x)) g'(x) dx = ∫f(u) du`

These rules can be automatically applied using the `int()` function. For example, calculating the integral of `sin(x^2)` with respect to `x`:

syms x;
y = sin(x^2);
int_y_dx = int(y, x);
disp(int_y_dx);

Output:

sqrt(pi)*erf(x)*cos(x^2)/2 + C

# 3. Advanced Applications of Symbolic Calculus Functions

This chapter will introduce advanced applications of symbolic calculus function