# 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.

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:

*x*

_{ 0 }and

*x*

_{ 0 }+ :

*f*(

*x*

_{ 0 }) and

*f*(

*x*

_{ 0 }+ ). Here means some small change of an argument, called an

*argument increment*; correspondingly a difference between the two values of a function:

*f*(

*x*

_{ 0 }+ ) –

*f*(

*x*

_{ 0 }

*) is called a*

*function increment*.

*Derivative*of a function

*y*=

*f*(

*x*) at a point

*x*

_{ 0 }is the limit:

If this limit exists, then a function

*f*(

*x*) is a

*differentiable*function at a point

*x*

_{ 0 }. 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*(

*x*

_{ 0 },

*f*(

*x*

_{ 0 }

*)). In general case an equation of a straight line with a slope*

*f*’(

*x*

_{ 0 }

*) has the shape:*

*y*=

*f*’(

*x*

_{ 0 }

*) ·*

*x + b .*

To find
*
b
*
we’ll
*
*
use the fact, that a tangent line goes through a point A :

*f*(

*x*

_{ 0 }

*) =*

*f*’(

*x*

_{ 0 }

*) ·*

*x*

_{ 0 }

*+ b*,

hence,
*
b
*
=
*
f
*
(
*
x
*
_{
0
}
*
*
) –
*
f
*
’(
*
x
*
_{
0
}
*
*
)
·
*
x
*
_{
0
}
*
*
, and substituting this expression instead of
*
b
*
, we’ll receive the equation of a tangent:

*y*=

*f*(

*x*

_{ 0 }

*) +*

*f*’(

*x*

_{ 0 }

*) · (*

*x – x*

_{ 0 }

*) .*

**
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

*t*

_{ 0 }till

*t*

_{ 0 }+

*the point displacement is equal to:*

*x*(

*t*

_{ 0 }+ ) –

*x*(

*t*

_{ 0 }) =

*, and its*

*average velocity*is:

*v*/

_{ a }=*.*As 0, then an average velocity value approaches the certain value, which is called an

*instantaneous velocity v*(

*t*

_{ 0 }

*) of a material point in the moment*

*t*

_{ 0 }. But according to the derivative definition we have:

hence,
*
v
*
(
*
t
*
_{
0
}
*
*
)
*
= x’
*
(
*
t
*
_{
0
}
*
*
), 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
*
).