2. Use the limit definition of the derivative to find the derivative $$ f^{\prime}(x) $$ of the function.

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To find the derivative $$ f^{\prime}(x) $$ of a function using the limit definition of the derivative, we use the formula:

$$ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Given that we are not provided with a specific function, let's assume the function is $$ f(x) = x^2 $$ for simplicity.

Substitute $$ f(x) = x^2 $$ into the formula:

$$ f^{\prime}(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} $$

Now, we can simplify the expression inside the limit and then calculate the limit to find the derivative $$ f^{\prime}(x) $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{h\rightarrow 0}\left(\frac{\left(x+h\right)^{2}-x^{2}}{h}\right)$$
- step1: Subtract the terms:
  $$\lim _{h\rightarrow 0}\left(\frac{2xh+h^{2}}{h}\right)$$
- step2: Reduce the fraction:
  $$\lim _{h\rightarrow 0}\left(2x+h\right)$$
- step3: Rewrite the expression:
  $$\lim _{h\rightarrow 0}\left(2x\right)+\lim _{h\rightarrow 0}\left(h\right)$$
- step4: Calculate:
  $$2x+0$$
- step5: Calculate:
  $$2x$$
The derivative of the function $$ f(x) = x^2 $$ using the limit definition of the derivative is $$ f^{\prime}(x) = 2x $$.
To find the derivative $$ f^{\prime}(x) $$ using the limit definition, we use: $$ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ Assuming $$ f(x) = x^2 $$, the derivative is $$ f^{\prime}(x) = 2x $$.

Use the limit definition of the derivative to find the derivative of the function.

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