mathematics//calculus//derivative

A derivative is the instantaneous rate of change of one quantity with respect to another, the limit of how much it changes over a small interval divided by that interval, and engineers use it for two jobs: to write down how a system evolves, and to ask how sensitive an output is to an input. A tank whose level goes from 1.00 m to 1.02 m in 4 s is rising at about 5 mm/s; shrink the 4 s towards zero and the ratio becomes the derivative of the level at that instant. Velocity is the derivative of position, acceleration the derivative of velocity, and when the variable is time the derivative is written with a dot, \(v=\dot p=dp/dt\). In general,

A derivative is the instantaneous rate of change of one quantity with respect to another, the limit of how much it changes over a small interval divided by that interval, and engineers use it for two jobs: to write down how a system evolves, and to ask how sensitive an output is to an input. A tank whose level goes from 1.00 m to 1.02 m in 4 s is rising at about 5 mm/s; shrink the 4 s towards zero and the ratio becomes the derivative of the level at that instant. Velocity is the derivative of position, acceleration the derivative of velocity, and when the variable is time the derivative is written with a dot, v=p˙=dp/dtv=\dot p=dp/dtv=p˙​=dp/dt. In general,

f′(a)=lim⁡h→0f(a+h)−f(a)h.f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.f′(a)=h→0lim​hf(a+h)−f(a)​.

The slope of the tangent at aaa: how many units of output per unit of input, near that point and only there.

Physics is written in derivatives. An ODE x˙=f(x,u,t)\dot x=f(x,u,t)x˙=f(x,u,t) gives the derivative of the state as a function of the state and the inputs, and a mechanical system brings second derivatives, mp¨+cp˙+kp=Fm\ddot p+c\dot p+kp=Fmp¨​+cp˙​+kp=F for the oscillator. Solving the model means undoing the derivative, which is an integral.

A derivative is a sensitivity. A centrifugal pump's power grows roughly as the cube of its speed (the affinity laws), so dP/P≈3 dn/ndP/P\approx3,dn/ndP/P≈3dn/n: raising the speed by 1 % costs about 3 % more power. The same question asked of many inputs at once gives the gradient, and of many outputs the Jacobian.

Differentiating a measured signal amplifies its noise, because a small difference of noisy samples is divided by a small sampling period. A 12-bit encoder has steps of 1.53 mrad; read at 1 kHz, a single step of quantization looks like a speed of 88 °/s. That is why the derivative action of a PID acts on a filtered measurement, and why a drone measures its angular rate directly with a gyroscope instead of differencing its angle.

The name covers several objects. The ordinary derivative has one input; the partial derivative moves one of several inputs and holds the others; the gradient and the Jacobian collect partials into a vector and a matrix. On a computer a derivative is approximated by finite differences or computed exactly by automatic differentiation.