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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 tangent line at $a$

(b)

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

(c)

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

(d)

the secant line at $a$

Note: this example question includes the introduction of a concept with corresponding short quiz questions. Read more here.

Correct answer

(a) the tangent line at $a$

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Question

QID:

demo/annotated/LectureVelocity/2-Derivative

Title:

Derivative definition

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Started at:

2026-08-04 23:50:11 (CDT)

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