From Secant to Tangent: The Derivative
The line through $(a, f(a))$ and $(a+h, f(a+h))$ is called a secant; its slope is $\dfrac{f(a+h) - f(a)}{h}$. The derivative is the limit of this slope as $h$ goes to zero: $f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$. The red line is the secant, the dashed green line the tangent.
What to notice
- As $h$ shrinks the secant settles onto the tangent; the sign of $h$ does not matter, it approaches the same line from the right and from the left.
- Choose $f(x) = |x|$ and take $a = 0$: for $h > 0$ the slope is always $1$, for $h < 0$ always $-1$. The two sides do not approach the same number, so the limit, and hence the derivative, does not exist.
- For $f(x) = e^x$ the slope of the tangent at every point equals the value of the function there.