Derivative definition

The instantaneous rate of change of the function $f(t)$ is given by its derivative.

Derivative of a function at a point

The derivative of $f(x)$ with respect to $x$ is the function $f'(x)$ defined as: $$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$ provided the limit exists.

Recall that

$$\frac{f(x+h) - f(x)}{h}$$

represents the slope of the secant line that connects points $(x,f(x))$ and $(x+h,f(x+h))$. Therefore, when we are computing the derivatives, we are taking the limit of a collection of slope lines when the interval $h$ is getting smaller. So what is the graphical representation of the derivative?

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The graphical interpretation for the derivative of $f$ with respect to $x$ evaluated at point $a$, i.e. $f'(a)$, is:

(a)

the secant line connecting $a$ and $a+h$

(b)

the secant line at $a$

(c)

the tangent line at $a$

(d)

the tangent line connecting $a$ and $a+h$

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demo/annotated/LectureVelocity/2-Derivative

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Derivative definition

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2026-08-04 22:47:36 (CDT)

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{
  "fprime": {
    "key": "c",
    "html": "the tangent line at $a$ ",
    "score": 1,
    "feedback": null
  }
}

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Derivative definition

Derivative of a function at a point

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