What does the derivative tell you about a function?

The derivative f'(x) tells you the instantaneous rate of change of the function at any point x. Geometrically, it is the slope of the tangent line to the curve. When f'(x) > 0, the function is increasing; when f'(x) < 0, it is decreasing; and when f'(x) = 0, the function has a horizontal tangent (potential maximum, minimum, or inflection point).

What is a tangent line?

A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. The equation of the tangent line at x=a is y = f(a) + f'(a)(x - a). It represents the best linear approximation to the function near that point.

Why is e^x special in calculus?

The function e^x is the only function (up to constant multiples) that is its own derivative: d/dx[e^x] = e^x. This makes it fundamental to calculus and differential equations. It appears in compound interest, population growth, radioactive decay, and many natural processes.

Derivative Visualizer

Q: What is a tangent line?

A tangent line is a straight line that touches a curve at exactly one point and has the same slope as the curve at that point. The equation of the tangent line at x=a is y = f(a) + f'(a)(x - a). It represents the best linear approximation to the function near that point.

Q: Why is e^x special in calculus?

The function e^x is the only function (up to constant multiples) that is its own derivative: d/dx[e^x] = e^x. This makes it fundamental to calculus and differential equations. It appears in compound interest, population growth, radioactive decay, and many natural processes.

Pick a function, drag the point along the curve, and watch the tangent line follow. Toggle the derivative curve to see f'(x) drawn alongside f(x).

Controls

Function

x² x³ sin(x) cos(x) eˣ ln(x) √x 1/x

x = 1.00

1.00

Show f'(x) curve

Values

f(x)	—
Slope	—
f'(x)	—

The Math Behind It

What Is a Derivative?

The derivative f'(x) is the slope of the tangent line at point x
It measures the instantaneous rate of change
Formally: f'(x) = lim_h→0 [f(x+h) - f(x)] / h
Positive slope = function increasing; negative = decreasing; zero = flat (potential extremum)

Common Derivatives

d/dx [x²] = 2x
d/dx [sin(x)] = cos(x)
d/dx [eˣ] = eˣ
d/dx [ln(x)] = 1/x

Try This

Select sin(x) and toggle Show f'(x) — you'll see cos(x) appear
Move x to where sin(x) peaks — the slope is exactly 0
Try x² — the tangent slope increases linearly (f'(x) = 2x)
Try eˣ — uniquely, the function equals its own derivative
Try 1/x — watch the tangent slope flip sign across the discontinuity at x=0

Key Insight

The derivative transforms a position question (“where am I?”) into a velocity question (“how fast am I changing?”). Every time you look at a speedometer, you're reading a derivative.

Derivative Visualizer

Controls

Values

The Math Behind It

What Is a Derivative?

Common Derivatives

Try This

Key Insight

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