using the first principles show that the derivative of f(x) = ax^2 +bx+c is 2ax+b
To find the derivative of the function f(x) = ax^2 + bx + c using first principles, we start by using the definition of the derivative:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Let's substitute f(x) = ax^2 + bx + c into this definition:
f'(x) = lim(h→0) [(a(x + h)^2 + b(x + h) + c) - (ax^2 + bx + c)] / h
Expanding the terms inside the brackets:
f'(x) = lim(h→0) [ax^2 + 2axh + ah^2 + bx + bh + c - ax^2 - bx - c] / h
Note that many terms cancel out:
f'(x) = lim(h→0) [2axh + ah^2 + bh] / h
Now, let's simplify further:
f'(x) = lim(h→0) (2axh/h + ah^2/h + bh/h)
Since h is approaching zero, we can simplify the expression:
f'(x) = lim(h→0) (2ax + ah + b)
Finally, taking the limit as h approaches zero:
f'(x) = 2ax + b
So, we have shown that the derivative of f(x) = ax^2 + bx + c using first principles is 2ax + b.