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 we’ll derive an equation of a tangent of a function graph at a point
A ( x0 , f ( 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 b we’ll use the fact, that a tangent line goes through a point A :
f ( x0
) = f ’( x0
) · x0
+ b ,
hence, b = f ( x0
) – f ’( x0
) · x0
, and substituting this expression instead of b , we’ll 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 time: a = v’ ( t ).
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