Concept preview

What you will be able to do

1. 01State the limit definition of the derivative at a point, $f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$, and its equivalent form $\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$.
2. 02Compute a derivative from first principles for polynomials such as $x^2$ and $x^3$, and translate between the notations $f'(x)$, $\dfrac{dy}{dx}$, $\dfrac{df}{dx}$, and $D_x f$.
3. 03Distinguish the derivative at a point (a number) from the derivative as a function, and use $f'(a)$ to write the equation of the tangent line.

The derivative of $f$ at the point $x=a$, written $f'(a)$, is defined as the limit of the average rates of change as the interval shrinks: $f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}.$ Read it slowly. Inside the limit is exactly the secant slope from 2.1 — the average rate of change over $[a,\,a+h]$. The $\lim_{h\to 0}$ out front performs the shrinking: it reports the value the secant slopes approach as the second point slides into the first. So $f'(a)$ is the instantaneous rate of change of $f$ at $a$, and geometrically it is the slope of the tangent line to $y=f(x)$ at $x=a$.