In algebra, you learned how to calculate the slope $(\Delta y/\Delta x)$ of a line between two points on a curve. However, the question of how to find the slope of a line tangent to a curve at a single point (see Figure 1) was left unanswered.

In calculus, one may approach the latter question by calculating the slope between two points $(c, f(c))$ and $(x, f(x))$ (Figure 2), and taking the limit as the points approach each other ( Figure 3). To wit,

$$\lim_{x \to c} \frac{\Delta y}{\Delta x} = \lim_{x\to c} \frac{f(x)-f(c)}{x-c}$$ It is often more convenient to rewrite this expression in terms of $\Delta x = x - c$. Hence, $$\lim_{x \to c} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x)-f(x)}{\Delta x}$$ This expression is one of the most important in calculus, so it has a special name: the derivative. The derivative is itself a function, and it is symbolized $f'(x)$. $$\boxed{f'(x) \equiv \lim_{\Delta x \to 0} \frac{f(x + \Delta x)-f(x)}{\Delta x}}$$ If $f$ is a function of more than one variable, it is important to indicate with respect to what variable the function is being differentiated. Therefore, several alternative notations have been devised: $$f'(x) = \frac{\diff f}{\diff x} = \frac{\diff}{\diff x} f(x) = D_x f(x)$$

For a function to be differentiable over an entire interval, it is required that the function be continuous over that interval: All differentiable functions are continuous. This is because all derivatives involve limits, and limits do not exist at discontinuities. However, the converse of this statement is not true: not all continuous functions are differentiable. There is even one function—the Weierstrass function—that is continuous everywhere but differentiable nowhere!