4.3 Application of differentiation

  Basic Mathematics and Measurements

Application of differentiation are,

(A). Slope of a curve
(B). Rate of change
(C). Maxima and minima
(D). Fractional and percentage changes

(A). Slope of a curve

$\frac{{dy}}{{dx}}$ gives the slope of a curve at a particular point.
$$\tan \theta = \frac{{dy}}{{dx}}$$

(B). Rate of change

$\frac{{dy}}{{dx}} = $ Rate of change of $y$ with respect to $x$

$\frac{{dx}}{{dt}} = $ Rate of change of position with respect to time $t$

$\frac{{dA}}{{dt}} = $ Rate of change of area $A$

$\frac{{dm}}{{dt}} = $ Rate of change of mass $m$

(C). Maxima and minima

It is an important application of differentiation as it helps us to find the maximum and minimum value of a function.

Let us consider a function $y=f(x)$ as shown in the figure.

It becomes minimum at $x_1$ and maximum at $x_2$. At these points, the tangent to the curve is parallel to the $x-$ axis and its slope is $\tan \theta = 0$

i.e. $$\frac{{dy}}{{dx}} = 0$$

At $\frac{{dy}}{{dx}} = 0$, it can either be minimum or maximum.

To confirm whether the function is minimum or maximum, we will find the second derivative.

For maximum value,

$$\frac{{dy}}{{dx}} = 0\quad {\text{and}}\quad \frac{{{d^2}y}}{{d{x^2}}} < 0$$

For minimum value,

$$\frac{{dy}}{{dx}} = 0\quad {\text{and}}\quad \frac{{{d^2}y}}{{d{x^2}}} > 0$$

(D). Fractional and percentage changes

If $y = k{A^a}{B^b}{C^c}$ where $k$, $m$ and $n$ are constants.

$$\Delta y\% = a\left( {\Delta A\% } \right) + b\left( {\Delta B\% } \right) + c\left( {\Delta C\% } \right)$$

The above equation is valid for small change and many number of variables.

Proof:

Let the function be,
$$y = k{A^a}{B^b}{C^c}$$
Taking $\log$ on both sides we get,
$$\log y = \log \left( {k{A^a}{B^b}{C^c}} \right)$$$$\log y = \log k + \log {A^a} + \log {B^b} + \log {C^c}$$$$\log y = \log k + a\log A + b\log B + c\log C$$Taking derivative both sides we get,
$$\frac{{dy}}{y} = a\left( {\frac{{dA}}{A}} \right) + b\left( {\frac{{dB}}{B}} \right) + c\left( {\frac{{dC}}{C}} \right)$$$$\frac{{dy}}{y} \times 100 = \left[ {a\left( {\frac{{dA}}{A}} \right) + b\left( {\frac{{dB}}{B}} \right) + c\left( {\frac{{dC}}{C}} \right)} \right] \times 100$$$$\Delta y\% = a\left( {\Delta A\% } \right) + b\left( {\Delta B\% } \right) + c\left( {\Delta C\% } \right)$$