## Differentiation

In linear motion the position of a moving object can be specified by a single position coordinate *x* at any time *t*.

The instantaneous velocity *v*_{x} of an object in linear motion is the rate of change of its position coordinate with respect to time.

In linear motion, a plot of *x* against *t* is called a position-time graph. The gradient of the tangent to such a graph at any particular value of *t* determines the instantaneous velocity of the moving object at that time. The velocity is constant if and only if the position-time graph is a straight line.

The instantaneous acceleration *a*_{x} of an object in linear motion is the rate of change of its velocity with respect to time.

In linear motion a plot of *v*_{x} against *t* is called a velocity-time graph. The gradient of the tangent to such a graph at any particular value of *t* determines the instantaneous acceleration of the moving object at that time. The acceleration is constant if and only if the velocity-time graph is a straight line.

If a single value of a dependent variable *x* can be associated with each value of an independent variable *t* in some specified domain, then we say that *x* is a function of *t*, and we speak of the function *x*(*t*).

Given a function *x*(*t*), the rate of change of *x* with respect to *t* at any particular value of *t* is given by,

*dx*/*dt* = lim(Δ*t* → 0)(Δ*x*/Δ*t*) = lim(Δ*t* → 0){[*x*(*t* + Δ*t*) – *x*(*t*)]/Δ*t*}

If a unique limit exists for all values of *t* in some domain, then this formula defines a function called the derivative or derived function,

*x*′(*t*) or *dx*/*dt*(*t*)

The value of the derivative *dx*/*dt* of a function *x*(*t*) at any given value of *t* is equal to the gradient of the tangent to the graph of *x* against *t* at that value of *t*.

In linear motion, the position coordinate of a moving object may be regarded as a function of *t* and written *x*(*t*). For such an object

*v*_{x}(*t*) = *dx*/*dt* = rate of change of position coordinate with respect to time

*a*_{x}(*t*) = *dv*_{x}/*dt* = rate of change of velocity with respect to time

Generally, if *y* is a function of *x*, the derivative *dy*/*dx* describes the rate of change of *y* with respect to x.

If *y*(*x*) is the linear function *y*(*x*) = *ax* + *b* then *dy*/*dx* = *a*

If *y*(*x*) is the quadratic function *y*(*x*) = *ax*^{2} + *bx* + *c* then *dy*/*dx* = 2*ax* + *b*