Application Of Differentiation

The differentiation is the subfield of Calculus and there are various Application of Differentiation in real world. The differentiation is very important part of Math as it is used in many scientific fields. Differentiation can be defined as the process of finding the Derivatives of the Functions. Differentiation can be used as a tool to calculate or study the rate of change of a quantity with respect to change in some other quantity. The most common example is calculation of velocity and acceleration. Velocity is given by v = dx / dt, where ' x ' is the distance covered by a moving body in time ‘t’.

Similarly acceleration can be given by can be given by a = dv / dt as acceleration is rate of change of velocity with respect to time. Here ' a ' is the acceleration ' v ' is the velocity and ' t ' is time.
 

Now we will see some other applications of differentiation-

1 ) Normal’s and tangents- Differentiation can be used to find the tangents and normal’s of curve we are studying the different forces acting on a body.

Tangent- Tangent can be defined as a straight line that touches the curve at a Point and the Slope of curve and line is same.

Normal- The perpendicular line to the Tangent of a curve is known as normal.

Slope can be calculated by using dy / dx or Slope= dy / dx.
 

2 ) Curvilinear motion- As we can calculate the velocity and acceleration of a moving body we can also use differentiation in curvilinear in which object moves along a curved path. Here we express x and y as function of time and it is known as parametric form. Here horizontal component of velocity is given by v_x = dx / dt, vertical component of velocity is given by v_y = dy / dt.

Magnitude is calculated by v = √ ( v²_x + v²_y ).

Direction ⊖ of an object can be calculated by tan ⊖_v = v_y / v_x
 

3 ) Related rates- When two are varying with respect to time and if they are related, then they can be expressed in terms of each other. We will have to differentiate both sides with respect to time d / dt.
 

4 ) Drawing a curve- We can sketch a curve using differentiation, we can find the Maxima and Minima using given data by finding the first derivative that is dy / dx or y ' and putting it equals to 0 that is y ' = 0 if value of ' x ' is positive then function has local minima and otherwise function has a local maxima. Then we calculate second derivative that is d² ( y ) / dx that is y' ' and if value of y' ' is greater than 0 or y' ' > 0 then curve has minimum type shape otherwise curve has a maximum type shape.