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.
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 vx = dx / dt, vertical component of velocity is given by vy = dy / dt.
Magnitude is calculated by v = √ ( v2x + v2y ).
Direction ⊖ of an object can be calculated by tan ⊖v = vy / vx
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.
Slope, length, area for polar curves in Calculus is one of the most interesting and bit complex topic of Calculus. In this section, we will look at areas enclosed by polar curves. Here, we used the...Read More
A secant line is defined as a Straight Line that passes through any two points lying on the curve. Let us assume that a function given as: Y = f(x), where 'Y' is dependent on independent variable 'x'. Thus a function specifies ...Read More
Slope in Derivatives is a simple and very useful concept in Calculus. We will learn here that how to find the Slope in the different type of Differential Equations by the help of derivative. We will go through the several ways for finding the Slope of a derivative and also solves some of the problems related to the evaluation of the slopes in the derivatives.
When we make an angle with the positive direction of x axis in anticlockwise sense is called as a tangent of line or Slope or gradient of line. So, tangent is a trigonometric angle, which is called as a Slope or gradient of the line. The slope of line is generally denoted by m, where
m = tan t, here t is the angle which makes with the positive direction of x axis in ant...Read More
Normal Differentiation is a method of obtaining the rate at which a dependent variable or a dependent output say ‘y’ changes with respect to the change in a independent variable or input. This rate of change is called the derivative of y with respect to x. However physical meaning of normal differentiation says that if a graph is plotted between a dependent var...Read More
In Geometry, a technique that defines basic concept of shape in a plane is called as curve sketching. So, curve sketching Calculus is basically used for solving a mathematical problem about shapes in geometry and for solving the typical mathematical problems like area, maximum, minimum value of certain equation or curve. For sketching a curve, we use following steps -