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Calculus: derivatives
Intermediate
The rate of change of one quantity with respect to another.
v = dx/dtA derivative is the instantaneous rate of change: the slope of the tangent to a graph. In physics, velocity is the derivative of position and acceleration the derivative of velocity. Useful rules: d(xⁿ)/dt = nxⁿ⁻¹, d(sin ωt)/dt = ω cos ωt.
If x = 5t², then v = 10t and a = 10 m/s².
A derivative needs the quantity as a function of the variable; derivative with respect to time differs from with respect to position.
In more detail
A derivative measures the rate of change of a quantity, equal to the slope of its graph at a point. Velocity is the derivative of position with respect to time.
Real-world uses
- Speedometers effectively measure the derivative of distance with respect to time.
- Economists use derivatives to measure how costs change with production, the same idea applied to money.
- A speedometer shows the instantaneous rate of change of position, which is the derivative of the distance with respect to time.
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Frequently asked questions
Why is the derivative useful in physics?
Many laws are written as rates of change, such as acceleration being the rate of change of velocity.
What is the derivative of x squared?
2x, which gives the slope of the parabola at any point x.
What is the derivative of a constant?
Zero, because a constant does not change. This is why the velocity of an object moving at a steady speed is constant.
Related topics in Measurement & Math Tools
Dot product · Cross product · Triple products · Calculus: integrals · Partial derivatives · Gradient, divergence, curl