Application of Derivatives

With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature. — Whitehead

6.1 Introduction

Uses of $\dfrac{dy}{dx}$: rates of change, increasing / decreasing functions, tangents and normals, maxima and minima, and approximations.

6.2 Rate of Change of Quantities

If $y=f(x)$, then $\dfrac{dy}{dx}$ is the rate of change of $y$ with respect to $x$.

If $x=f(t)$ and $y=g(t)$, then (chain rule)

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\qquad \frac{dx}{dt}\mathrel{\htmlClass{course-neq}{\char"2260}} 0$

$\dfrac{dy}{dx}>0$ if $y$ increases as $x$ increases; $\dfrac{dy}{dx}<0$ if $y$ decreases as $x$ increases.

6.3 Increasing and Decreasing Functions

On an interval $I$:

• Increasing: $x_1<x_2\Rightarrow f(x_1)<f(x_2)$ (equivalently $f'(x)\ge 0$ in the usual NCERT test)
• Decreasing: $x_1<x_2\Rightarrow f(x_1)>f(x_2)$ (equivalently $f'(x)\le 0$)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$:

• $f'(x)>0$ on $(a,b)$ $\Rightarrow$ $f$ increasing on $[a,b]$
• $f'(x)<0$ on $(a,b)$ $\Rightarrow$ $f$ decreasing on $[a,b]$
• $f'(x)=0$ on $(a,b)$ $\Rightarrow$ $f$ constant

To find intervals: solve $f'(x)=0$, split the real line at those points, and check the sign of $f'$.

6.4 Maxima and Minima

A critical point $c$ is where $f'(c)=0$ or $f$ is not differentiable.

First derivative test

As $x$ increases through $c$:

• $f'$ changes $+$ to $-$ $\Rightarrow$ local maximum
• $f'$ changes $-$ to $+$ $\Rightarrow$ local minimum
• $f'$ does not change sign $\Rightarrow$ point of inflexion

Second derivative test

Assume $f''(c)$ exists.

• $f'(c)=0$ and $f''(c)<0$ $\Rightarrow$ local maximum
• $f'(c)=0$ and $f''(c)>0$ $\Rightarrow$ local minimum
• $f'(c)=f''(c)=0$ $\Rightarrow$ test fails; go back to the first derivative test

Absolute maxima / minima on $[a,b]$

1. Find critical points in $(a,b)$
2. Evaluate $f$ at those points and at $a$, $b$
3. The largest value is the absolute maximum; the smallest is the absolute minimum

Chapter Summary

• Rate: $\dfrac{dy}{dx}$; with a parameter $t$, use $\dfrac{dy/dt}{dx/dt}$
• Increasing when $f'>0$; decreasing when $f'<0$
• Critical points: $f'=0$ or not differentiable
• Second test: $f''<0$ max, $f''>0$ min
• Closed interval: check endpoints as well as critical points

Exercises (NCERT)

• Exercise 6.1 — rates of change
• Later exercises — increasing/decreasing, maxima/minima
• Miscellaneous Exercise