Understanding the Derivative of 2x^2 - x by the First Principle

Understanding calculus, specifically the concept of the derivative, is a fundamental part of higher mathematics. One of the critical methods to understand derivatives is the first principle (also known as the limit definition of a derivative). In this tutorial, we'll go through the process of finding the derivative of f(x) 2x^2 - x using the first principle. This step-by-step explanation will help you grasp the underlying concepts and gain confidence in your calculus skills.

Step-by-Step Derivation

To find the derivative of f(x) 2x^2 - x using the first principle, we need to use the definition:

f'(x) limh→0 [f(x h) - f(x)] / h.

Step 1: Calculate f(x h)

First, we need to find f(x h) for the given function:

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

Expanding (x h)^2 gives:

(x h)^2 x^2 2xh h^2

Substituting this back into the function:

f(x h) 2(x^2 2xh h^2) - (x h) 2x^2 4xh 2h^2 - x - h

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

Now we find f(x h) - f(x):

f(x h) - f(x) [2x^2 4xh 2h^2 - x - h] - [2x^2 - x]

Simplifying the expression:

f(x h) - f(x) 4xh 2h^2 - h

Step 3: Divide by h

Next, we divide the above expression by h:

(f(x h) - f(x)) / h (4xh 2h^2 - h) / h

Simplifying further:

(4xh 2h^2 - h) / h 4x 2h - 1

Step 4: Take the Limit as h Approaches 0

Finally, we take the limit as h approaches 0:

limh→0 (4x 2h - 1) 4x - 1

Final Result

After following these steps, we find that the derivative of f(x) 2x^2 - x is:

f'(x) 4x - 1

Conclusion

This process illustrates the power and utility of the first principle in finding derivatives. It may seem cumbersome initially, but mastering the first principle is crucial for more advanced calculus topics. Bootstrap your understanding and build confidence in your calculus skills with this method.

Additional Resources

If you're looking to deepen your understanding of derivatives and the first principle, consider exploring the following resources:

MIT OpenCourseWare - Single Variable Calculus Khan Academy - Derivatives Paul's Online Math Notes - Calculus I