Differentiating the Function f(x) 3x^2 - 2 from First Principles

Differential calculus plays a crucial role in understanding the behavior of functions by analyzing the rate at which they change. This article delves into the process of differentiating the function f(x) 3x2 - 2 from the ground up, using the foundational concept of the derivative from first principles. This approach is not only a fundamental tool in calculus but also enhances our comprehension of how derivatives are derived.

Understanding the Derivative from First Principles

The derivative of a function f(x) is defined as:

df(x)/dx lim_{h to 0} frac{f(x h) - f(x)}{h}

Step-by-Step Differentiation of f(x) 3x^2 - 2

Step 1: Calculate f(x h)

First, we need to find f(x h):

f(x h) 3(x h)^2 - 2

Expanding this:

f(x h) 3(x^2 2xh h^2) - 2 3x^2 6xh 3h^2 - 2

Step 2: Calculate f(x h) - f(x)

Next, we need to find f(x h) - f(x):

f(x h) - f(x) (3x^2 6xh 3h^2 - 2) - (3x^2 - 2)

This simplifies to:

6xh 3h^2

Step 3: Divide by h

Now, divide by h:

frac{6xh 3h^2}{h} 6x 3h

Step 4: Take the Limit as h Approaches 0

Finally, we take the limit as h to 0:

lim_{h to 0} (6x 3h) 6x

Thus, the derivative of f(x) 3x^2 - 2 with respect to x is:

f'(x) 6x

Final Result

The derivative of f(x) 3x^2 - 2 with respect to x is:

f'(x) 6x