How to find derivative using limit definition

Understand the Problem

The question is asking how to find a derivative using the limit definition, which involves calculating the limit of the difference quotient as the increment approaches zero.

Answer

The derivative of the function $f(x) = x^2$ is $f'(x) = 2x$.

Steps to Solve

Define the Function and Increment Let the function be $f(x)$ and the increment be $h$. The difference quotient is given by the formula:

$$ \frac{f(x+h) - f(x)}{h} $$

Set Up the Limit Expression To find the derivative using the limit definition, we set up the limit as $h$ approaches zero:

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

Substitute the Function Substitute the specific function you want to differentiate into the limit expression. For example, if $f(x) = x^2$, substitute it into the formula:

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

Simplify the Expression Expand and simplify the expression inside the limit:

$$ = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} $$

This simplifies to:

$$ = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

Factor and Cancel Factor out $h$ from the numerator:

$$ = \lim_{h \to 0} (2x + h) $$

Evaluate the Limit Now, evaluate the limit as $h$ approaches zero:

$$ f'(x) = 2x $$

The derivative of the function $f(x) = x^2$ is $f'(x) = 2x$.

More Information

This method of finding the derivative is based on the limit definition of a derivative, which provides a foundational understanding of how derivatives describe the rate of change of a function.

Tips

Not using the limit properly: Some students forget to take the limit as $h \to 0$ after simplifying the difference quotient. Always remember that the limit is critical in defining the derivative.
Incorrectly simplifying the expression: Be careful during algebraic manipulation. Ensure every term is accounted for when expanding and simplifying.

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