Why do we need differentiation? There are many applications of differentiation in the fields of engineering and science especially when modelling the behaviour of moving objects. The need for differentiation is found in analysis of finance and economics. One of the important applications of differentiation is in the area of optimisation, which means finding the condition for a maximum (or minimum) to occur. This is important in business where we have to minimise the cost and maximise the profit and in engineering where we have to maximise the strength and minimise the cost. Differentiation is needed to solve many types of real-world problems.
Differentiation or Derivative as a rate of change :
Let y = f(x) be a function of x. Let `Delta` y be the change in y corresponding to a small change `Delta`x in x. Then, `(Deltay)/(Deltax)` represents the change in y due to a unit change in x. In other words, `(Deltay)/(Deltax)` represents the average rate of change of y w.r.t. x as x changes from x to x+`Delta`x.As `Delta`x`->` 0, the limiting value of this average rate of change of y with respect to x in the interval [x,x+`Delta` x] becomes the instantaneous rate of change of y w.r.t. x.Thus,
`lim_(Deltax->0)` `(Deltay)/(Deltax)` = Instantaneous rate of change of y w.r.t. x
`rArr` `dy/dx` = Rate of change of y w.r.t. x [ because `lim_(Deltax->0)` `(Deltay)/(Deltax)` = `dy/dx` ]
Hence, `dy/dx` represents the rate of change of y w.r.t. x for a definite value of x. We need differentiation when rate change is not constant. Lets consider an example:
Example: The total cost C(x) associated with the production of x units of an item is given by
C(x)=0.005x3 -0.02x2 +30x+5000
Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.
Sol: Since the marginal cost is the rate of change of total cost with respect to the output. Therefore,
Marginal cost (MC) = `(dC)/dx` (x) = `d/dx` (0.005x3 - 0.02x2 + 30x + 5000)
= 0.005(3x2) -0.02(2x) + 30
when x=3, we have
Marginal cost (MC) = 0.05 * 3 * 32 - 0.02 * 2 * 3 + 30
= 0.135 - 0.12 + 30 = 30.015
Hence required marginal cost is Rs 30.02(approximately).
Applications of differentiation:
The need for differentiation is found in physics also. We need to find tangents and normals to curves when we are modelling real world problems in physics like forces acting on a body moving in circular path. We can find the gradient of a tangent or normal at any point (x,y) using differentiation. For example,
If we tie a stone to one end of a thread and rotate it in circular motion and then let it go, the path followed by the stone is tangential to the circle of the motion.
Similarly, if a satellite, revolving around the earth in a circular motion, under the influence of gravity, is suddenly thrown out from the orbit of the earth, it follows a path tangential to the circle.
Likewise, if we are travelling in a car around a corner and we hit something slippery on the road (like oil, ice or loose gravel) and our car starts to skid, it will continue in a direction tangent to the curve.
Example : Find the slope of the tangent to the curve x2+3y+y2=5 at (1,1).
Sol:Differentiating the curve with respect to x, we get
2x + 3`dy/dx` + 2y`dy/dx` = 0
`rArr` `dy/dx` = - `(2x)/(2y+3)`
`rArr` `(dy/dx)` (1,1) = - `(2)/(2+3)` = -`(2)/(5)`
Therefore the slope of tangent at (1,1) = `(dy/dx)` (1,1) = `(-2)/(5)`
In physics we can find the velocity of a body from the displacement function using v = `(ds)/dt`
and acceleration from the velocity function a = `(dv)/(dt)` = `(d^2s)/(dt^2)`
My forthcoming post is on Line Segments and class 10 syllabus cbse will give you more understanding about Algebra.
Importance and need for differentiation:
The study of differentiation is helpful in the calculation of small changes (or errors) in dependent variables corresponding to small changes (or errors) in the independent variable and is of great importance in the theory of errors in Engineering, Physics, Statistics and several other branches of the science. We can calculate absolute error, relative error and percentage error. We can find maximum or minimum values of a function in its domain. In real world, we are trying to do things like maximise the profit in a company, or minimise the cost or find the least amount of material to make a particular object. These are very important in the world of industry. We can draw a complete circle through points on part of a curve. The radius of curvature is the radius of approximating circle. We can find the radius of curvature by the knowledge of differentiation. We can also find the local maxima, local minima, the point of inflexion and sketch the curve using differentiation.
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