How does the derivative measure change at a specific point?

The derivative is a fundamental concept in calculus that measures how much a function's output changes in response to a change in its input, at a specific point. In other words, it quantifies the rate at which a function is changing at a particular instant.

Mathematically, the derivative of a function f(x) at a point x=a is defined as the limit of the difference quotient as the change in x approaches zero:

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

Here, f'(a) is the derivative of f at the point x=a, and h is an incremental change in x. As h approaches zero, the difference quotient approaches the slope of the tangent line to the curve y=f(x) at the point (a, f(a)). This slope represents the rate of change of the function at that specific point.

For example, consider the function f(x) = x². To find the derivative at a specific point, say x=3, we calculate:

$f'(3) = \lim_{h \to 0} \frac{(3+h)² - 3²}{h} = \lim_{h \to 0} \frac{9 + 6h + h² - 9}{h} = \lim_{h \to 0} (6 + h) = 6$

So, the derivative of f(x) = x² at x=3 is 6, which means the function is changing at a rate of 6 units per unit change in x at that point.