To differentiate [tex]\( y = x^2 \)[/tex] from first principles, we follow the process of using the difference quotient for the derivative. The difference quotient is defined as:
[tex]\[ \frac{y(x + h) - y(x)}{h} \][/tex]
where [tex]\( h \)[/tex] is a small increment.
Here are the detailed steps:
1. Define the function [tex]\( y = x^2 \)[/tex]:
The function is [tex]\( y = x^2 \)[/tex].
2. Substitute [tex]\( x + h \)[/tex] into the function:
Find [tex]\( y(x + h) \)[/tex]:
[tex]\[
y(x + h) = (x + h)^2 = x^2 + 2xh + h^2
\][/tex]
3. Set up the difference quotient:
The difference quotient is:
[tex]\[
\frac{y(x + h) - y(x)}{h} = \frac{(x^2 + 2xh + h^2) - x^2}{h}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h}
\][/tex]
4. Simplify the expression:
Factor out [tex]\( h \)[/tex] from the numerator:
[tex]\[
\frac{h(2x + h)}{h}
\][/tex]
Cancel [tex]\( h \)[/tex]:
[tex]\[
2x + h
\][/tex]
5. Take the limit as [tex]\( h \)[/tex] approaches 0:
To find the derivative, we take the limit of the simplified difference quotient as [tex]\( h \)[/tex] approaches 0:
[tex]\[
\lim_{h \to 0} (2x + h) = 2x
\][/tex]
Therefore, the derivative of [tex]\( y = x^2 \)[/tex] from first principles is:
[tex]\[ \boxed{2x} \][/tex]
However, given the numerical approximation, if we consider a very small value for [tex]\( h \)[/tex] such as [tex]\( h = 1 \times 10^{-5} \)[/tex]:
1. First, calculate the difference quotient using [tex]\( h = 1 \times 10^{-5} \)[/tex]:
[tex]\[
\frac{(x + 1 \times 10^{-5})^2 - x^2}{1 \times 10^{-5}}
\][/tex]
2. Expand [tex]\((x + 1 \times 10^{-5})^2\)[/tex]:
[tex]\[
= x^2 + 2x \times 1 \times 10^{-5} + (1 \times 10^{-5})^2
\][/tex]
Therefore:
[tex]\[
y(x + h) = x^2 + 2x \times 10^{-5} + 1 \times 10^{-10}
\][/tex]
3. Plug this into the difference quotient:
[tex]\[
\frac{x^2 + 2x \times 10^{-5} + 1 \times 10^{-10} - x^2}{1 \times 10^{-5}}
\][/tex]
4. Simplify the numerator:
[tex]\[
\frac{2x \times 10^{-5} + 1 \times 10^{-10}}{1 \times 10^{-5}}
\][/tex]
Divide each term by [tex]\( 1 \times 10^{-5} \)[/tex]:
[tex]\[
2x + 1 \times 10^{-10} / 1 \times 10^{-5} = 2x + 1 \times 10^{-5}
\][/tex]
Therefore, the numerical result is:
[tex]\[ 2x + 1 \times 10^{-5} \][/tex]
Thus, the derivative is:
[tex]\[ \boxed{2x + 1 \times 10^{-5}}. \][/tex]
The expression [tex]\( -100000.0 \cdot x^2 + 100000.0 \cdot (x + 1.0e-5)^2 \)[/tex] represents the unsimplified form of the difference quotient. The simplified form, after dividing by [tex]\( h \)[/tex] and taking the limit, gives us [tex]\( 2x + 1.0e-5 \)[/tex].
