Derivative of the Function

A derivative measures the rate at which a function changes with respect to its independent variable. For a function f: X -> Y, the derivative can be calculated as follows:

In simple terms, the derivative of a function represents how much that function is changing at any given point. It gives us the rate at which the function's value is changing with respect to its input (usually denoted as x)

We can calculate the derivative of the function using Python sympy library:


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import sympy as sp

# Create symbolic variable `x`
x = sp.symbols('x')
# Define the function
f = x**3 - 2*x**2 - 1

df_dx = sp.diff(f, x)  # Calculate the derivative

# Evaluate the derivatives at specific point
point = {x:2}
df_dx_value = df_dx.subs(point)

print(f'Derivative with respect to x: {df_dx}')
print(f'Derivative in point {point} is : {df_dx_value}')

Let's describe the code above:

Firstly, we use x = sp.symbols('x') to create a symbolic variable x which represents the argument of the function;
sp.diff(function, variable) method is used to calculate the function df_dx that represents the derivative of the function f;
Finally, we use df_dx.subs(point) method to calculate the value of the derivative in point x=2. The point variable is a dictionary where we store variable names and values.

How can the derivative be interpreted in real-life tasks?

The derivative has different interpretations in different fields of science. Let's name some of them:
Here are the mathematical, physical, and geometric interpretations of derivatives:

Mathematical interpretation: The derivative of a function at a point represents the rate of change of the function at that point. It provides information about how the function is changing, whether it is increasing or decreasing, and the steepness of the curve at that particular point:
- if and only if the derivative is positive, then the function increases at the chosen point;
- if and only if the derivative is negative, then the function decreases at the chosen point;
- if and only if the derivative equals zero, then the function doesn't change at the chosen point.

Note

All of the above statements are true only for functions for which the derivative exists at any point in the domain.

Physical interpretation: In physics, the derivative has various interpretations depending on the context. For example:
- Velocity: The derivative of the position function with respect to time gives the velocity, which represents the rate of change of position with respect to time.
- Acceleration: The derivative of the velocity function with respect to time gives the acceleration, which represents the rate of change of velocity with respect to time.

Geometric interpretation: The derivative also has a geometric interpretation of tangent lines and slopes. At any point on a curve, the derivative gives the slope of the tangent line to the curve at that point. The slope of the tangent line indicates the rate at which the curve changes at that specific point.

These interpretations highlight derivatives' fundamental concepts and applications in mathematics, physics, and geometry, providing insights into the behavior of functions and physical phenomena.

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Section 3. Chapter 1