3.5 Notation and Differentiability

Notes

There are several ways to write a derivative.

• $f'(x)$, $\displaystyle \frac{dy}{dx}$, $\displaystyle \frac{d}{dx}[f(x)]$ and $y'$ all mean the derivative.
• $\displaystyle \frac{dy}{dx}$ is read "the derivative of $y$ with respect to $x$". It is a single symbol, not a fraction to cancel.
• $f$ is differentiable at $a$ if $f'(a)$ exists. Corners, cusps, vertical tangents and discontinuities make a function non-differentiable.
• If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The reverse is false: $f(x) = |x|$ is continuous at 0 but not differentiable there.