Derivatives and connected rates
Change you can see.
Explore limits, derivatives, and accumulation through motion, measurement, and design.
Follow the real situation, derive the mathematics, and test it with interactive graphs and worked practice.
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Derivatives and connected rates
Watch a secant become a tangent.
Two position observations give average velocity. Bringing their times closer reveals instantaneous velocity.
Explore the idea ↗Interactive lessonUse derivative rules to measure sensitivity.
A nonlinear response can become much more sensitive as its input grows. Compare its value with its local slope.
Explore the idea ↗Interactive lessonChanging one dimension can change another.
A fixed-perimeter rectangle grows one side while shrinking the other. Its area depends on both changes.
Explore the idea ↗Interactive lessonA ratio responds to numerator and denominator changes.
A rectangle's aspect ratio can change sharply as one side narrows, even when total perimeter remains fixed.
Explore the idea ↗Interactive lessonAn oscillation's slope has its own phase.
A vibrating mass stops momentarily at a turning point, yet moves fastest near the center. Follow displacement and velocity together.
Explore the idea ↗Interactive lessonA rate can pass through two transformations.
A sensor first rescales time-dependent input, then cubes the response. Follow each stage to understand its final rate.
Explore the idea ↗Interactive lessonA constraint determines the allowed slope.
A particle constrained to a circular guide cannot change its horizontal position independently of its height.
Explore the idea ↗Interactive lessonAn angle changes at a nonlinear rate.
A viewing angle responds less and less to the same extra horizontal displacement as the geometry spreads out.
Explore the idea ↗Interactive lessonA proportional growth rate makes an exponential.
When growth is proportional to the amount already present, equal time intervals multiply rather than add the same quantity.
Explore the idea ↗Interactive lessonPosition, velocity, and acceleration tell different stories.
A cart can move backward while accelerating forward. The signs of its velocity and acceleration determine whether its speed rises or falls.
Explore the idea ↗Interactive lessonThe same inflow can raise a cone's level at different rates.
A conical tank widens as it fills. Change its depth and flow to see why level rise slows even with a constant inlet.
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