Differentiation: Definition and Fundamental Properties
What you must understand
The derivative is a limit of a difference quotient: $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$, and at a point $a$ it is the slope of the tangent line. Interpret it as instantaneous rate of change. Know differentiability implies continuity (the converse fails, e.g., $|x|$ at $0$).
Master the power, constant, sum, product, and quotient rules, plus derivatives of the elementary functions ($\sin x$, $\cos x$, $e^x$, $\ln x$). Understand tangent-line equations, normal lines, and linear approximation. Recognize notation variants ($f'$, $\frac{dy}{dx}$, $y'$).
Traps
- Product rule errors: differentiating only one factor instead of $f'g+fg'$.
- Quotient rule sign and order mistakes; the denominator is squared.
- Confusing the derivative value with the function value when writing tangent lines.
- Assuming continuity guarantees differentiability (corners, cusps, vertical tangents).
- Misreading difference-quotient limits that disguise a derivative at a shifted point.
Worked example
Find the tangent line to $f(x)=x^3-2x$ at $x=1$. $f(1)=1-2=-1$. $f'(x)=3x^2-2$, so $f'(1)=1$. Tangent: $y-(-1)=1(x-1)$, i.e., $y=x-2$.
Rule of thumb: the derivative at a point gives both the slope and the local linearization; always compute $f(a)$ and $f'(a)$ separately.