We have analyzed the concept of derivative/Differentiation of a function, this concept is widely used in various fields like math, physics, engineering sciences, etc, derivatives are used to analyze any function by calculating its first and second derivative, Hence we can see that there are various applications of derivatives in various fields so it is very important to study these applications of derivatives which we’re discussing here.
Some of the applications which we’re going to discuss are –
- Derivative as rate measurer
- Equation of tangent and normal to a curve
- Monotonicity and Extrema (maxima and minima) of functions
- Approximations using differentials
Differentiation as rate measurer
While knowing differentiation/derivatives for the very first time we defined two meanings of derivative of a function one is the Geometrical meaning and the other one is Physical meaning.
The Derivative/Differentiation of a function f(x) with respect to x at x=a, Physically means as the instantaneous rate of change of f(x) w.r.t x. Hence the slope at a given point on a graph gives an instantaneous rate of change. This physical interpretation of finding instantaneous rate of change is widely used in Various Disciplines whether it’s Physics, Chemistry, Accountancy, etc. for example- in physics, we find Instantaneous Velocity by differentiating displacement w.r.t time.
Monotonicity
the meaning of monotonicity in the context of a function refers to whether a function is increasing or decreasing, any function can be analyzed as increasing or decreasing by looking at its derivatives.
A function is defined to be Monotonically increasing if [Latexpage]
$$ f(x_{2}) \ge f(x_{1}) \hspace{3mm} \forall \hspace{3mm} x_{2} \ge x_{1} $$
| Note – equality should hold at discrete points in the above expression then the function will be monotonic increasing. Similarly, we can state for monotonic decreasing. |
For the function to be Monotonically increasing, the first derivative should always be greater than equal to zero.
$$ f'(x) \ge 0 $$
It should be noted that the equality should hold at discrete points otherwise the function can be a constant function, which is called both monotonically increasing and decreasing, similarly for Monotonically decreasing function the first derivate should be less than equal to zero i.e.
$$ f'(x) \le 0 $$
Extrema (Maxima or Minima)
Finding the maximum and minimum values of functions around a particular set of domains that is local maxima or local minima, and the overall maxima and minima that are Global maxima/ Global minima of functions is very important to study as it’s applied in various disciplines of study.
Derivatives are used for finding the extremum (means maximum or minimum) value of various functions over various domains of study. generally, we use the first time differentiation(first derivative) and its successive differentiation that is the second derivative to find maxima and minima.
First Derivative Test
The first derivative test is based on the use of the concept that for an increasing function the first derivative is positive and for decreasing its negative as we studied in monotonicity. before going any further in the first derivative test, we would like to introduce the meaning of critical points and Stationary Points in mathematics
Critical Points and Stationary Points
for any function y = f(x), the values of x co-ordinate where the first derivative is zero, or where the function is non-differentiable, i.e. f'(x) [first derivative] doesn’t exist are called the Critical points,
and The values of x coordinate where the first derivative equals zero are called Stationary points. One can see that all stationary points are critical points but the reverse statement need not be true. Now, let’s see some of the curves and analyze the behavior of the first derivative around the stationary points
from above one can see that whenever the sign of the first derivative changes from positive to negative there occurs a local maximum, where function attains maximum value in its neighborhood.
similarly, when the sign of the first derivative changes from negative to positive there occurs a local minimum, where the function attains minimum value in its neighborhood,
One should note that the above discussion only talks about locally maxima and minima and doesn’t ensure that this maximum, minimum is globally (overall) maxima for function over its domain of definition.
Keeping above in mind the global maxima and minima can be found as-
Talking for y = f(x) in interval $ x \epsilon [a,b] $ which is differentiable
- Max{f(x)}= $Max{ f(a), f(b), f(c_{1}),f(c_{2})…}$
- Min{f(x)}= $Min{ f(a), f(b), f(c_{1}),f(c_{2})…}$
Here a, b are endpoints of the interval, $c_{1},c_{2}, …$ are critical points within domain of study.
Second Derivative Test
Besides the first derivative test, we can also analyze the maxima and minima of function by thinking about its second derivative.
Talking for y = f(x) in interval $ x \epsilon [a,b] $ which is twice differentiable [the second derivative exists] the following things can be deduced by thinking over the statements given here.
- for any point x=c in given interval if $f'(c)=0 \hspace{2mm} and \hspace{2mm} f”(c)>0$ then at x=c there exists local minima.
- for any point x=c in given interval if $f'(c)=0 \hspace{2mm} and \hspace{2mm} f”(c)
it should be noted that if $f'(c)=0 \hspace{2mm} and \hspace{2mm} f”(c)=$ then if we analyze, we can’t conclude the maxima and minima hence the second derivative analysis can’t be used here, and one has to analyze the first derivative.
Equation of tangent and Normal
Tangent
The Geometrical Meaning of the derivative of a function at a point x=a means the slope of the tangent at that point. Geometrically derivative of a function f(x) at a point x=a means the slope of the tangent at x=a.
$$
f'(x)= \frac{d}{dx}f(x) =\ tan{\theta}= slope
$$
the principle to calculate the slope of tangent using derivatives is the first principle of differentiation one can see detailed proof. the slope of the tangent is given by-
$$ f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} $$
since we get the slope of the tangent at any point as the first derivative of a function at that point, so we can find the equation of tangent also at that point by using the two-point form of the equation of a straight line,
$$y–y_{1} = m(x – x_{1})$$Hence the equation of tangent to any curve passing through point $ P(x_{1},y_{1}) $is given as
$$ y–y_{1} = f'(x_{1})(x–x_{1}) $$Normal
The equation of the normal at any point on a curve can be found by knowing the slope of the tangent at that point, as we know the product of the slope of two perpendicular lines is always negative 1.
$$ m_{1}.m_{2}= -1 $$as f'(x) is slope of tangent , hence slope of normal will be -1/f'(x_{1}) , Hence equation of normal at any point $ P(x_{1},y_{1})$ on a curve is given as
$$ y–y_{1} = \frac{-1}{f'(x_{1}})(x–x_{1}) $$Approximations
derivatives are also used to make certain approximations, the one which we are dealing with is deduced from the definition of derivative/differentiation.
$$ f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} $$
from above things can be approximated as
$$ f(x+h)=h.f'(x)+f(x)$$
Stay tuned with Laws of Nature for more useful and interesting content.
