Derivative As Rate Measure

If a quantity y depends on and varies with a quantity x, then the rate of change of y with respect to x is dy/dx.

For example, the rate of change of pressure p with respect to height h is dp/dh.

A rate of change with respect to time is simply called as the rate of change. A rate of change of current i is di/dt and a rate of change of temperature θ is dθ/dt.

Velocity and Acceleration

A car describes a distance x metres in time t seconds along a straight road. If the velocity v is constant, then

v = x/t 

The slope (gradient) of the distance-time graph is constant.

If the velocity of the car is not constant then the distance-time graph is not be a straight line.

The velocity of the car at any instant is given by gradient of the distance-time graph. If an expression for the distance x is known in terms of time, then the velocity is obtained by differentiating the expression.

v = dx/dt

The acceleration a of the car is defined as the rate of change of velocity.

a = dv/dt

The acceleration of the car at any instant is given by the gradient of the velocity-time graph. If an expression for velocity is known in terms of time t, then the acceleration is obtained by differentiating the expression.

The acceleration is given by the second differential coefficient of distance x with respect to time t.

a = d²x/dt²