What is the derivative of a function?

The derivative of a function is a fundamental concept in calculus that measures how much the output of the function changes in response to a change in its input, at a specific point. In other words, it's the rate at which the function is changing at an exact location. This concept is crucial for finding the slope of a tangent to a curve at a specific point, as you've been practicing.

Given a function y = f(x), the derivative of f with respect to x, denoted as f'(x) or dy/dx, is a new function that gives the slope of the tangent to the curve y = f(x) at any point x. It's calculated using the limit of a difference quotient:

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

Here's a simple example: If you have the function f(x) = x², then its derivative f'(x) is calculated as follows:

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

So, the derivative of f(x) = x² is f'(x) = 2x, which means the slope of the tangent to the curve y = x² at any point x is 2x.