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Derivative. Geometrical and mechanical meaning of derivative

Derivative. Argument and function increments. Differentiable function.
Geometrical meaning of derivative. Slope of a tangent. Tangent equation.
Mechanical meaning of derivative. Instantaneous velocity. Acceleration.

Derivative. Consider a function y = f ( x ) at two points:  x0  and  x0 + f ( x0 ) and  f ( x0 + ). Here   means some small change of an argument, called an argument increment; correspondingly a difference between the two values of a function: f ( x0 + ) –  f ( x0 )  is called a  function increment. Derivative of a function  y = f ( x ) at a point  x0 is the limit:

If  this limit exists, then a function  f ( x ) is a differentiable function at a point  x0. Derivative of a function  f ( x ) is marked as:

Geometrical meaning of derivative.  Consider a graph of a function  y = f ( x ) :

From  Fig.1  we see, that for any two points A and B of the function graph:

where  - a slope angle of the secant AB.
So, the difference quotient is equal to a secant slope. If to fix the point A and to move the point B towards A, then  will unboundedly decrease and approach 0, and the secant AB will approach the tangent AC. Hence, a limit of  the difference quotient is equal to a slope of a tangent at point A. Hence it follows: a derivative of a function at a point is a slope of a tangent of this function graph at this point.

Tangent equation. Now well derive an equation of a tangent of a function graph at a point  A(x0f(x0 )). In general case an equation of a straight line with a slope  f ( x0 )  has the shape:

y = f ( x0 ) x + b .

To find  well use the fact, that a tangent line goes through a point A :

f ( x0 ) = f ( x0 ) x0 + b ,

hence,  bf ( x0 ) – f ( x0 ) x0  ,  and substituting this expression instead of  b , well receive the equation of a tangent:

y =  f ( x0 ) +  f ( x0 ) ( x – x0  ) .

Mechanical meaning of derivative.  Consider the simplest case: a movement of a material point along a coordinate line, moreover, the motion law is given,  i.e. a coordinate  x  of this moving point is the known function  x ( t )  of  time  t. During the time interval from  t0  till  t0 +  the point displacement is equal to: x (t0 + ) –  x ( t0 ) = ,  and its average velocity is:  va = / . As  0, then an average velocity value approaches the certain value, which is called an  instantaneous velocity  v ( t0 ) of a material point in the moment  t0 . But according to the derivative definition we have:

hence,  v ( t0 ) = x ( t0 ), i.e. a derivative of a coordinate with respect to time is a velocity. This is a mechanical meaning of a derivative. Analogously to this, an acceleration is a derivative of a velocity with respect to timea = v ( t ).


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