Định nghĩa Đạo hàm của Hàm số và Sử dụng Ký hiệu Đạo hàm

The derivative: instantaneous rate, packaged

• The instantaneous rate of change is so useful it gets its own name: the derivative 导数.
• It turns "slope of the tangent at each $x$" into a brand-new function, $f'(x)$.
• Feed it an $x$, and it returns the slope of $f$ right there.
• This lesson makes the definition precise and shows the notations you'll use everywhere.

The limit definition

• The derivative of $f$ is
• $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
• Inside is a difference quotient — the secant slope over an interval of width $h$.
• Taking $h\to0$ shrinks the interval to nothing, turning the secant slope into the tangent slope.

Secant slopes approach the tangent slope: the derivative is the limit of average rates

Secant slopes approach the tangent slope as h shrinks to 0

The "at a point" version

• To find the slope at one specific input $a$, use the alternate form:
• $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
• Here $x$ slides toward $a$ instead of a step $h$ shrinking to $0$ — same idea, different bookkeeping.
• Both definitions are just "secant slope, in the limit," so they always agree.

Reading the notations

• The same derivative wears several outfits — learn to recognize all of them:
• Lagrange: $f'(x)$ (read "$f$ prime of $x$").
• Leibniz: $\dfrac{dy}{dx}$ or $\dfrac{d}{dx}[f(x)]$ — emphasizes "rate of $y$ with respect to $x$."
• All mean the instantaneous rate of change, i.e. the slope of the tangent line at $x$.