Syllabus:
\[\text{Limits of polynomials and rational functions – Limits of the form}\ \lim\ _{x\ \to\ 0}\ \frac{sin ax}{bx}\ and\ \hspace{2cm}\\ \lim\ _{x\ \to\ 0}\ \frac{tan ax}{bx} \text{(x in radions) (results only) – Definition of differentiability -}\ \hspace{2cm}\\ \text{Differentiation formulae for standard functions – Differentiation of sum, difference, product and}\\ \text{quotient of functions – chain rule – second order derviaties}\]
\[\color {royalblue} {\text{Limt of the function f(x)}}:\ \hspace{20cm}\]
\[When\ the\ variable\ x\ approaches\ a\ (constant)\ and\ if\ the\ function\ f ( x )\ approaches\ a\ constant\ l,\]
\[then\ l\ is\ called\ the\ limit\ value\ of\ f ( x )\ as\ x\ approaches\ a\ and\ is\ denoted\ as\ \lim\ _{x\ \to\ a}\ f(x)\ =\ l.\]
\[\color {royalblue} {Properties}:\ \hspace{20cm}\]
\[1)\ \lim\ _{x\ \to\ a}\ [f(x)\ \pm\ g(x)]\ =\ lim\ _{x\ \to\ a}\ f(x)\ \pm\ lm\ _{x\ \to\ a}\ g(x)\]
\[2)\ \lim\ _{x\ \to\ a}\ [K\ f(x)\ ]\ =\ K\ \lim\ _{x\ \to\ a}\ f(x)\]
\[3)\ \lim\ _{x\ \to\ a}\ [f(x)\ .\ g(x)]\ =\ lim\ _{x\ \to\ a}\ f(x)\ . \ lm\ _{x\ \to\ a}\ g(x)\]
\[4)\ \lim\ _{x\ \to\ a}\ [\frac{f(x)}{g(x)}]\ =\ \frac{\lim\ _{x\ \to\ a}\ f(x)}{\ lm\ _{x\ \to\ a}\ g(x)}\ provided\ \ lm\ _{x\ \to\ a}\ g(x)\ \neq\ 0.\]
\[\color {royalblue} {\text{LIMITS OF POLYNOMIALS}}:\ \hspace{10cm}\]
\[\color {purple} {Example\ 1:}\ \color {red} {Evaluate:}\ Lt\ _{x\ \to\ 3}\ (x^2\ -\ 5x\ +\ 2)\ \hspace{10cm}\]
\[\color {blue}{Solution:}\ \hspace{20cm}\]
\[Lt\ _{x\ \to\ 3}\ (x^2\ -\ 5x\ +\ 2)\ =\ (3)^2\ -\ 5(3)\ +\ 2\ \hspace{10cm}\]
\[=\ 9\ -\ 15\ +\ 2\ \hspace{3cm}\]
\[=\ -\ 4\ \hspace{4cm}\]
\[=\ -\ 4\ \hspace{5cm}\]
\[\boxed{Lt\ _{x\ \to\ 3}\ (x^2\ -\ 5x\ +\ 2)\ =\ -\ 4}\ \hspace{7cm}\]
\[\color {royalblue} {\text{LIMITS OF RATIONAL FUNCTIONS}}:\ \hspace{10cm}\]
\[\color {purple} {Example\ 2.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 2}\ \frac{x^2\ +\ 5}{x^2\ +\ 3}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ \hspace{20cm}\]
\[\lim\ _{x\ \to\ 2}\ \frac{x^2\ +\ 5}{x^2\ +\ 3}\ =\ \frac{\lim\ _{x\ \to\ 2}\ (x^2\ +\ 5)}{\ lm\ _{x\ \to\ 2}\ (x + 3)}\ =\ \frac{2^2\ +\ 5}{2\ +\ 3}\ =\ \frac{9}{5}\ \hspace{8cm}\]
DIFFERENTIATION
\[\color {royalblue} {Definition}:\ \hspace{20cm}\]
\[Suppose\ f\ is\ a\ real\ valued\ function,\ the\ function\ defined\ by\ \lim\ _{h\ \to\ 0}\ \frac{f(x\ +\ h)\ -\ f(x)}{x\ -\ h}\]\[is\ defined\ to\ be\ the\ derivative\ of\ f\ at\ x\ and\ is\ denoted\ by\ f\prime(x).\]\[f\prime(x)\ is\ denoted\ by\ \frac{d}{dx}\ (f(x))\ or\ \frac{dy}{dx}\ (where\ y\ =\ f(x))\]
\[\color {royalblue} {Formulae}:\ \hspace{20cm}\]
\[1.\ \frac{d}{dx}\ (x^n)\ =\ n\ x^{n\ -\ 1}\]
\[2.\ \frac{d}{dx}\ (\sqrt{x})\ =\ \frac{1}{2\ \sqrt{x}}\]
\[3.\ \frac{d}{dx}\ (e^x)\ =\ e^x\]
\[4.\ \frac{d}{dx}\ (log\ x)\ =\ \frac{1}{x}\]
\[5.\ \frac{d}{dx}\ (Sin\ x)\ =\ Cos\ x\]
\[6.\ \frac{d}{dx}\ (Cos\ x)\ =\ -\ Sin\ x\]
\[7.\ \frac{d}{dx}\ (Tan\ x)\ =\ Sec^2\ x\]
\[8.\ \frac{d}{dx}\ (Cot\ x)\ =\ -\ Cosec^2\ x\]
\[9.\ \frac{d}{dx}\ (Sec\ x)\ =\ Sec\ x\ Tan\ x\]
\[10.\ \frac{d}{dx}\ (Cosec\ x)\ =\ -\ Cosec\ x\ Cot\ x\]
\[11.\ \frac{d}{dx}\ (a^x)\ =\ a^x\ log\ a\]
\[12.\ \frac{d}{dx}\ (Sin^{-1}\ x)\ =\ \frac{1}{\sqrt{1\ -\ x^2}}\]
\[13.\ \frac{d}{dx}\ (Cos^{-1}\ x)\ =\ -\ \frac{1}{\sqrt{1\ -\ x^2}}\]
\[14.\ \frac{d}{dx}\ (Tan^{-1}\ x)\ =\ \frac{1}{1\ +\ x^2}\]
\[\color {royalblue} {Properties}:\ \hspace{20cm}\]
\[1.\ If\ u\ and\ v\ are\ functions\ of\ x,\ Then\ \frac{d}{dx}\ (u\ \pm\ v)\ =\ \frac{du}{dx}\ \pm\ \frac{dv}{dx}\]
\[2.\ If\ u\ is\ a\ function\ of\ x,\ and\ k\ is\ a\ constant,\ Then\ \frac{d}{dx}\ (ku)\ =\ k \frac{du}{dx}\]
\[3.\ If\ k\ is\ any\ constant,\ Then\ \frac{d}{dx}\ (k)\ =\ 0\]
\[4.\ If\ u\ and\ v\ are\ functions\ of\ x,\ Then\ \frac{d}{dx}\ (u\ v)\ =\ u\ \frac{dv}{dx}\ +\ v\ \frac{du}{dx}\]
\[4.\ If\ u\ v\ and\ w\ are\ functions\ of\ x,\ Then\ \frac{d}{dx}\ (u\ v\ w)\ =\ u\ v\ \frac{dw}{dx}\ +\ v\ w\ \frac{du}{dx}\ +\ w\ u\ \frac{dv}{dx}\]
\[4.\ If\ u\ and\ v\ are\ functions\ of\ x,\ Then\ \frac{d}{dx}\ (\frac{u}{v})\ =\ \frac{v\ \frac{du}{dx}\ -\ u\ \frac{dv}{dx}}{v^2}\]
\[\color {purple} {Example\ 5:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ x\ +\ x^2\ +\ x^3\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ x\ +\ x^2\ +\ x^3\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ \frac{d}{dx}(x)\ +\ \frac{d}{dx}(x^2)\ +\ \frac{d}{dx}(x^3)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 1\ +\ 2\ x\ +\ 3\ x^2\ \hspace{10cm}\]
\[\color {purple} {Example\ 6:}\ \color {Red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ 3\ x^2\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ 3\ x^2\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ 3\ \frac{d}{dx}(x^2)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 3\ (2x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 6\ x\ \hspace{10cm}\]
\[\color {purple} {Example\ 7:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ 3\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ 3\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ \frac{d}{dx}(3)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 0\ \hspace{10cm}\]
\[\color {purple} {Example\ 8:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ sin\ x\ +\ tan\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ sin\ x\ +\ tan\ x\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ \frac{d}{dx}(sin\ x)\ +\ \frac{d}{dx}(tan\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ cos\ x\ +\ sec^2\ x\ \hspace{10cm}\]
\[\color {purple} {Example\ 9:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ x^3\ sin\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ x^3\ sin\ x\ \hspace{15cm}\]
\[Here\ u\ =\ x^3,\ \hspace{5cm}\ v\ =\ sin\ x\]
\[W.\ K.\ T\ \frac{d}{dx}\ (u\ v)\ =\ u\ \frac{dv}{dx}\ +\ v\ \frac{du}{dx}\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^3\ \frac{d}{dx}(sin\ x)\ +\ sin\ x\ \frac{d}{dx}(x^3)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^3\ cos\ x\ +\ sin\ x\ (3\ x^2)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^3\ cos\ x\ +\ 3\ x^2\ sin\ x\ \hspace{10cm}\]
\[\color {purple} {Example\ 10:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ e^x\ log\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ e^x\ log\ x\ \hspace{15cm}\]
\[Here\ u\ =\ e^x,\ \hspace{5cm}\ v\ =\ log\ x\]
\[W.\ K.\ T\ \frac{d}{dx}\ (u\ v)\ =\ u\ \frac{dv}{dx}\ +\ v\ \frac{du}{dx}\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ e^x\ \frac{d}{dx}(log\ x)\ +\ log\ x\ \frac{d}{dx}(e^x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ e^x\ \frac{1}{x}\ +\ log\ x\ e^x\ \hspace{10cm}\]
\[\color {purple} {Example\ 11:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ x^2\ sin\ x\ log\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ x^2\ sin\ x\ log\ x\ \hspace{15cm}\]
\[Here\ u\ =\ x^2,\ \hspace{2cm}\ v\ =\ sin\ x\ \hspace{2cm}\ w\ =\ log\ x\]
\[W.\ K.\ T\ \frac{d}{dx}\ (u\ v\ w)\ =\ u\ v\ \frac{dw}{dx}\ +\ v\ w\ \frac{du}{dx}\ +\ w\ u\ \frac{dv}{dx}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ x^2\ Sin\ x\ \frac{d}{dx}\ (log\ x)\ +\ sin\ x\ log\ x\ \frac{d}{dx}\ (x^2)\ +\ log\ x\ x^2\ \frac{d}{dx}\ (sin\ x)\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ x^2\ sin\ x\ (\frac{1}{x})\ +\ sin\ x\ log\ x\ 2\ x\ +\ log\ x\ x^2\ Cos\ x\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ x^3\ Sin\ x\ (Sec^2\ x)\ +\ 3\ Sin\ x\ Tan\ x\ x^2\ +\ x^3\ Tan\ x\ Cos\ x\ \hspace{10cm}\]
\[\color {purple} {Example\ 12:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ \frac{x\ +\ 3}{x\ -\ 3}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ \frac{x\ +\ 3}{x\ -\ 3}\ \hspace{15cm}\]
\[Here\ u\ =\ (x\ +\ 3),\ \hspace{5cm}\ v\ =\ (x\ -\ 3)\]
\[W.\ K.\ T\ \frac{d}{dx}\ (\frac{u}{v})\ =\ \frac{v\ \frac{du}{dx}\ -\ u\ \frac{dv}{dx}}{v^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{(x\ -\ 3)\ \frac{d}{dx}\ (x\ +\ 3)\ -\ (x\ +\ 3)\ \frac{d}{dx}\ (x\ -\ 3)}{(x\ -\ 3)^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{(x\ -\ 3)\ (1\ +\ 0)\ -\ (x\ +\ 3)\ (1\ -\ 0)}{(x\ -\ 3)^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{x\ -\ 3\ -\ x\ -\ 3}{(x\ -\ 3)^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{-\ 6}{(x\ -\ 3)^2}\ \hspace{10cm}\]
\[\color {purple} {Example\ 13:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ \frac{ax\ +\ b}{cx\ +\ d}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ \frac{ax\ +\ b}{cx\ +\ d}\ \hspace{15cm}\]
\[Here\ u\ =\ (ax\ +\ b),\ \hspace{5cm}\ v\ =\ (cx\ +\ d)\]
\[W.\ K.\ T\ \frac{d}{dx}\ (\frac{u}{v})\ =\ \frac{v\ \frac{du}{dx}\ -\ u\ \frac{dv}{dx}}{v^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{(cx\ +\ d)\ \frac{d}{dx}\ (ax\ +\ b)\ -\ (ax\ +\ b)\ \frac{d}{dx}\ (cx\ +\ d)}{(cx\ +\ d)^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{(cx\ +\ d)\ (a\ +\ 0)\ -\ (ax\ +\ b)\ (c\ +\ 0)}{(cx\ +\ d)^2}\ \hspace{10cm}\]
\[ \frac{dy}{dx}\ =\ \frac{ad\ -\ bc}{(cx\ +\ d)^2}\ \hspace{10cm}\]
Sometimes y is not defined directly as a function of x but is given as a function of another variable, say ‘u’ which is defined as a function of x. Hence y is indirectly a function of x. In such case y is said to be a function of function.
\[\color {royalblue} {Chain Rule}:\ \hspace{20cm}\]
\[If\ ‘y’\ is\ a\ function\ of\ ‘u’\ and\ ‘u’\ is\ a\ function\ of\ ‘x’,\ Then\ \frac{dy}{dx}\ =\ \frac{dy}{du}\ ×\ \frac{du}{dx}\]
\[\color {purple} {Example\ 16:}\ \color {red} {Find\ \frac{dy}{dx}}\ if\ y\ =\ (3\ x\ +\ 6)^5\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ (3\ x\ +\ 6)^5\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ 5\ (3\ x\ +\ 6)^{5\ -\ 1}\ \frac{d}{dx}(3\ x\ +\ 6)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 5\ (3\ x\ +\ 6)^4\ 3\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 15\ (3\ x\ +\ 6)^4\ \hspace{10cm}\]
\[\color {purple} {Example\ 17:}\ \color {red} {Find\ \frac{d}{dx}}\ {cos(log\ x)}\ \hspace{15cm}\]
\[\frac{d}{dx}\ {cos(log\ x)}\ =\ -sin(log\ x)\ \frac{d}{dx}(log\ x)\ \hspace{10cm}\]
\[ =\ -\ sin(log\ x)\ \frac{1}{x}\ \hspace{10cm}\]
\[ =\ \frac{-sin(log\ x)}{x}\ \hspace{10cm}\]
\[\color {purple} {Example\ 18:}\ \color {red} {find\ \frac{dy}{dx}},\ if\ y\ =\ log(Sec\ x)\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ log(Sec\ x)\ \hspace{15cm}\]
\[\frac{dy}{dx}\ =\ \frac{1}{Sec\ x}\ \frac{d}{dx}(Sec\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ \frac{1}{Sec\ x}\ (Sec\ x\ Tan\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ tan\ x\ \hspace{10cm}\]
Successive Differentiation (Up to Second Order):
\[\text{If ‘y’ is a function of x, then differentiation of y gives}\ \frac{dy}{dx}.\ Again\ differentiation\ of\ \hspace{10cm}\\ \frac{dy}{dx}\ \text{w.r.t.x gives a new function of x which is called the second}\ \hspace{10cm}\\ \text{differential coefficient of y w.r.t.x and is denoted by the symbol}\ \frac{d^2y}{dx^2}\ \hspace{10cm}\]
\[\color {royalblue} {Notation\ of\ successive\ derivatives}:\ \hspace{20cm}\]
\[1.\ \frac{dy}{dx}\ =\ y_1\ =\ f^\prime(x)\ =\ D(y)\]
\[2.\ \frac{d^2y}{dx^2}\ =\ y_2\ =\ f^{\prime \prime}(x)\ =\ D^2(y)\]
\[\color {purple} {Example\ 19:}\ \color {red} {Find\ \frac{d^2y}{dx^2}}\ if\ y\ =\ tan\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ tan\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}(tan\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ Sec^2\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}( Sec^2\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 2\ sec\ x\ sec\ x\ tan\ x\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 2\ Sec^2\ x\ tan\ x\ \hspace{10cm}\]
\[\color {purple} {Example\ 20:}\ \color {red} {Find\ \frac{d^2y}{dx^2}}\ if\ y\ =\ Sin\ 3\ x\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ y\ =\ Sin\ 3\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( Sin\ 3\ x)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ Cos\ 3\ x\ \frac{d}{dx}(3\ x)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ Cos\ 3\ x\ 3(1)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ 3\ Cos\ 3\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}( 3\ Cos\ 3\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 3\ \frac{d}{dx}(Cos\ 3\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 3\ (-\ Sin\ 3\ x\ 3(1))\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ -\ 9\ Sin\ 3\ x\ \hspace{10cm}\]
