Derivation of a Function

Yaser Rahmati | یاسر رحمتی

Given a function $f(x)$, the derivative $f′(x)$ (read as "f prime of x") at a particular point $x$ is defined as the limit of the difference quotient as the interval around $x$ shrinks to zero:

$$lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

Example

Let's find the derivative of the function $f(x)=x^3$ using the definition of the derivative:

$$\acute{f(x)}=lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

Step 1:

Write down the function.

$$f(x)=x^3$$

Step 2:

Compute $f(x+h)$.

$$f(x+h)=(x+h)^3$$

Step 3:

Expand $(x+h)^3$ using binomial expansion or the distributive property.

$$f(x+h)=x^3+3x^2h+3xh^2+h^3$$

Step 4:

Subtract $f(x)$ from $f(x+h)$.

$$f(x+h)-f(x) = 3x^2h+3xh^2+h^3$$

Step 5:

Plug $f(x+h)−f(x)$ into the definition of the derivative.

$$\acute{f(x)}=lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=lim_{h\to 0}\frac{3x^2h+3xh^2+h^3}{h} = lim_{h\to 0}(3x^2+3xh+h^2)$$

Step 6:

Plug in $h=0$ to evaluate the limit.

$$f(x)=3x^2$$

Python

You can use the SymPy library in Python to symbolically calculate the derivative of a function. Here's how you can do it:

import sympy as sp

# Define the symbol and the function
x = sp.symbols('x')
f = x**3  # Define your function here

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

# Print the derivative
print("The derivative of f(x) =", f_prime)

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