Definition
The derivative of a function measures how the function's output changes as its input changes. Formally, the derivative of a function $f(x)$ at a point $x$ is defined as:
$$ f'(x) = lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$
This expression gives the instantaneous rate of change or the slope of the tangent line to the curve at the point $x$.
Worked Example
Find the derivative of $f(x) = x^2$.
Step 1: Write the definition.
$$ f'(x) = lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} $$
Step 2: Expand $(x+h)^2$.
$$ (x+h)^2 = x^2 + 2xh + h^2 $$
So,
$$ f'(x) = lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} $$
Step 3: Simplify the numerator.
$$ = lim_{h \to 0} \frac{2xh + h^2}{h} $$
Step 4: Factor $h$ from the numerator.
$$ = lim_{h \to 0} \frac{h(2x + h)}{h} $$
Step 5: Cancel $h$ and take the limit.
$$ = lim_{h \to 0} (2x + h) = 2x $$
Final Answer: The derivative is $f'(x) = 2x$.
Takeaways
- The derivative represents the instantaneous rate of change of a function.
- Use the limit definition to find derivatives from first principles.
- For $f(x) = x^2$, the derivative is $f'(x) = 2x$.