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 vx 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 ax of an object in linear motion is the rate of change of its velocity with respect to time.

In linear motion a plot of vx 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(Δt → 0)(Δxt) = lim(Δt → 0){[x(t + Δt) – x(t)]/Δt}

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

vx(t) = dx/dt = rate of change of position coordinate with respect to time

ax(t) = dvx/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) = ax2 + bx + c then dy/dx = 2ax + b

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~ by jamesdow2013 on April 15, 2013.

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