To find the derivative of the function [tex]\( f(x) = x^2 \)[/tex] using the first principle (also known as the limit definition of the derivative), we follow these steps:
1. Definition of the Derivative:
The derivative of [tex]\( f(x) \)[/tex] at a point [tex]\( x \)[/tex] is given by:
[tex]\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\][/tex]
2. Substitute the Function [tex]\( f(x) = x^2 \)[/tex]:
Now, we substitute [tex]\( f(x) = x^2 \)[/tex] into the definition:
[tex]\[
f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}
\][/tex]
3. Expand the Expression:
We need to expand [tex]\( (x+h)^2 \)[/tex]:
[tex]\[
(x+h)^2 = x^2 + 2xh + h^2
\][/tex]
Therefore, the expression inside the limit becomes:
[tex]\[
\frac{(x^2 + 2xh + h^2) - x^2}{h}
\][/tex]
4. Simplify the Numerator:
In the numerator, [tex]\( x^2 \)[/tex] and [tex]\( -x^2 \)[/tex] cancel each other out:
[tex]\[
\frac{2xh + h^2}{h}
\][/tex]
5. Factor out [tex]\( h \)[/tex] in the Numerator:
[tex]\[
\frac{h(2x + h)}{h}
\][/tex]
We can cancel [tex]\( h \)[/tex] in the numerator and the denominator (as long as [tex]\( h \neq 0 \)[/tex]):
[tex]\[
2x + h
\][/tex]
6. Take the Limit as [tex]\( h \)[/tex] Approaches 0:
Now, we take the limit of [tex]\( 2x + h \)[/tex] as [tex]\( h \)[/tex] approaches 0:
[tex]\[
\lim_{h \to 0} (2x + h) = 2x
\][/tex]
So, the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) = x^2 \)[/tex] using the first principle is:
[tex]\[
f'(x) = 2x
\][/tex]
