# 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: *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* ). Back |