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 $f'(x) = 2x$.
Answer for screen readers
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.
