Derivative. Geometrical and mechanical meaning of derivativeDerivative. 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 ).