Where is the derivative of sin(x) equal to zero?

Derivative is zero where the original function has a maximum or minimum.

What is the shape of the integral of a constant function?

Accumulating a constant rate gives a linear increase.

Sketch a function whose derivative is always positive but decreasing.

Increasing but at a slowing rate — like a logarithm.

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Pro 🔒~20 min

Calculus Grapher

Visualize derivatives and integrals graphically

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How it works

Calculus Grapher demonstrates a key principle: The derivative of a function gives the instantaneous rate of change — the slope of the tangent line at each point. The derivative of a function gives the instantaneous rate of change — the slope of the tangent line at each point. The integral gives the accumulated area under the curve. The Fundamental Theorem of Calculus connects them: differentiation and integration are inverse operations. These concepts underlie all of physics, from velocity/acceleration to electric flux.

Step-by-step

Select a preset function or draw your own.
The derivative curve (blue) shows slope; the integral curve (green) shows accumulated area.
Drag the x-position marker to read exact values.
Observe how a peak in f becomes a zero in f'.

Key formulas

$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ Definition of derivative
$\int_a^b f(x)\,dx$ Definite integral (area)
$\frac{d}{dx}[\sin x] = \cos x$ Example derivative

Frequently asked questions

Where is the derivative of sin(x) equal to zero?: Derivative is zero where the original function has a maximum or minimum.
What is the shape of the integral of a constant function?: Accumulating a constant rate gives a linear increase.
Sketch a function whose derivative is always positive but decreasing.: Increasing but at a slowing rate — like a logarithm.