The slope of the tangent at every $x$ at once is a new function, the derivative of $f$, written $f'(x)$ and read aloud f prime of x. Its definition is the last lesson's limit with $x$ in place of the fixed point $a$: $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$, read aloud: f prime of x equals the limit, as h approaches zero, of f of x plus h minus f of x, all over h. Here $f$ is the function, $x$ is any input, $h$ is the gap to a second input, shrinking to zero, and $f'(x)$ is the slope of the tangent at $x$. Working a derivative this way is called first principles, and the exam asks for it by name.

For $f(x) = x^2$: $f(x + h) = (x + h)^2 = x^2 + 2xh + h^2$. Subtract $f(x) = x^2$ and the numerator is $2xh + h^2$; every term has an $h$, so dividing by $h$ leaves $2x + h$; let $h \to 0$ and $f'(x) = 2x$. At $x = 3$ the slope is 6. The whole method lives in the first step, and so does the trap: $f(x + h)$ means the function applied to $x + h$; the $h$ goes inside and gets squared along with everything else. Writing $f(x + h)$ as $f(x) + h$ collapses the numerator to a lone $h$, the quotient to 1, and every function in the world to a slope of 1, which is nonsense and is easy to catch.