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Definition

The partial differential coefficient of $f(x,\,y)$ with respect to $x$ is the ordinary differential coefficient of $f(x,\,y)$ when $y$ is regarded as a constant. It is written as $\text{ }\frac{\partial f}{\partial x}$ or ${{D}_{x}}\,f$ or ${{f}_{x}}$.     Thus, $\frac{\partial f}{\partial x}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(x+h,\,y)-f(x,y)}{h}$     Again, the partial differential coefficient $\frac{\partial f}{\partial y}$ of $f(x,\,y)$ with respect to $y$ is the ordinary differential coefficient of $f(x,\,y)$ when x is regarded as a constant.     Thus, $\frac{\partial f}{\partial y}=\underset{k\to 0}{\mathop{\lim }}\,\frac{f(x,\,y+k)-f(x,\,y)}{k}$     e.g., If $z=f(x,\,y)={{x}^{4}}+{{y}^{4}}+3x{{y}^{2}}+{{x}^{2}}y+x+2y$     Then $\frac{\partial z}{\partial x}$ or $\frac{\partial f}{\partial x}$ or ${{f}_{x}}=4{{x}^{3}}+3{{y}^{2}}+2xy+1$ (Here $y$ is regarded as constant)               $\frac{\partial z}{\partial y}\ \ \text{or}\ \,\frac{\partial f}{\partial y}$ or ${{f}_{y}}=4{{y}^{3}}+6xy+{{x}^{2}}+2$(Here $x$ is regarded as constant)

Leibnitz?s Theorem

G.W. Leibnitz, a German mathematician gave a method for evaluating the ${{n}^{th}}$ differential coefficient of the product of two functions. This method is known as Leibnitz’s theorem.     Statement of the theorem : If $u$ and $v$ are two functions of $x$ such that their ${{n}^{th}}$ derivative exist then ${{D}^{n}}(u.v.)=$$^{n}{{C}_{0}}({{D}^{n}}u)v{{+}^{n}}{{C}_{1}}{{D}^{n-1}}u.Dv{{+}^{n}}{{C}_{2}}{{D}^{n-2}}u.{{D}^{2}}v+...........$${{+}^{n}}{{C}_{r}}{{D}^{n-r}}u.{{D}^{r}}v+.........+u.({{D}^{n}}v).$     The success in finding the ${{n}^{th}}$ derivative by this theorem lies in the proper selection of first and second function. Here first function should be selected whose ${{n}^{th}}$ derivative can be found by standard formulae. Second function should be such that on successive differentiation, at some stage, it becomes zero so that we need not to write further terms.

Differentiation of Integral Function

If ${{g}_{1}}(x)$ and ${{g}_{2}}(x)$ both functions are defined on $[a,\,\,b]$ and differentiable at a point $x\in (a,b)$ and $f(t)$ is continuous for ${{g}_{1}}(a)\le f(t)\le {{g}_{2}}(b)$, then     $\frac{d}{dx}\int_{{{g}_{1}}(x)}^{{{g}_{2}}(x)}{f(t)dt}=f[{{g}_{2}}(x)]{{{g}'}_{2}}(x)-f[{{g}_{1}}(x)]{{{g}'}_{1}}(x)$       $=f[{{g}_{2}}(x)]\frac{d}{dx}{{g}_{2}}(x)-f[{{g}_{1}}(x)]\frac{d}{dx}{{g}_{1}}(x)$.

${{n}^{th}}$ Derivative Using Partial Fractions

For finding ${{n}^{th}}$ derivative of fractional expressions whose numerator and denominator are rational algebraic expression, firstly we resolve them into partial fractions and then we find ${{n}^{th}}$ derivative by using the formula giving the ${{n}^{th}}$ derivative of $\frac{1}{ax+b}$.

Successive Differentiation or Higher Order Derivatives

(1) Definition and notation : If $y$ is a function of $x$ and is differentiable with respect to $x,$ then its derivative $\frac{dy}{dx}$can be found which is known as derivative of first order. If the first derivative $\frac{dy}{dx}$ is also a differentiable function, then it can be further differentiated with respect to x and this derivative is denoted by ${{d}^{2}}y/d{{x}^{2}}$, which is called the second derivative of $y$ with respect to $x$. Further if $\frac{{{d}^{2}}y}{d{{x}^{2}}}$is also differentiable then its derivative is called third derivative of $y$ which is denoted by $\frac{{{d}^{3}}y}{d{{x}^{3}}}$. Similarly ${{n}^{th}}$ derivative of $y$ is denoted by $\frac{{{d}^{n}}y}{d{{x}^{n}}}$. All these derivatives are called as successive derivatives and this process is known as successive differentiation. We also use the following symbols for the successive derivatives of $g(x)$ :     ${{y}_{1}},\,\,\,\,{{y}_{2}},\,\,\,\,{{y}_{3,}}.........,{{y}_{n}},......$              ${y}',\,\,\,\,{y}'',\,\,\,\,{y}'''.........,{{y}^{n}},......$     $Dy,\,\,\,\,\,{{D}^{2}}y,\,\,\,\,{{D}^{3}}y.........,{{D}^{n}}y,......$,     (where $D=\frac{d}{dx}$)          $\frac{dy}{dx},\,\,\,\,\frac{{{d}^{2}}y}{d{{x}^{2}}},\,\,\,\,\frac{{{d}^{3}}y}{d{{x}^{3}}},\,.......\,\,\,\frac{{{d}^{n}}y}{d{{x}^{n}}},...........$     ${f}'(x),\,\,\,\,{f}''(x),\,\,\,\,{f}'''(x),.........,{{f}^{n}}(x),......$     If $y=f(x)$, then the value of the ${{n}^{th}}$ order derivative at $x=a$ is usually denoted by  ${{\left( \frac{{{d}^{n}}y}{d{{x}^{n}}} \right)}_{x=a}}$ or ${{({{y}_{n}})}_{x=a}}$ or ${{({{y}^{n}})}_{x=a}}$ or ${{f}^{n}}(a)$       (2) ${{n}^{th}}$ Derivatives of some standard functions :       (I) (a) $\frac{{{d}^{n}}}{d{{x}^{n}}}\sin (ax+b)={{a}^{n}}\sin \left( \frac{n\pi }{2}+ax+b \right)$       (b)  $\frac{{{d}^{n}}}{d{{x}^{n}}}\cos (ax+b)={{a}^{n}}\cos \left( \frac{n\pi }{2}+ax+b \right)$       (II) $\frac{{{d}^{n}}}{d{{x}^{n}}}{{(ax+b)}^{m}}=\frac{m\,!}{(m-n)\,!}{{a}^{n}}{{(ax+b)}^{m-n}},$  where $m>n$     Particular cases       (i) When $m=n;$ ${{D}^{n}}\{{{(ax+b)}^{n}}\}={{a}^{n}}.n\,!$       (ii) (a) When $a=1,b=0$, then $y={{x}^{m}}$       $\therefore$${{D}^{n}}({{x}^{m}})=m(m-1).......(m-n+1){{x}^{m-n}}=\frac{m!}{(m-n)!}{{x}^{m-n}}$       (b) When $m<n,\,{{D}^{n}}\{{{(ax+b)}^{m}}\}=0$       (iii) When $a=1,\,b=0$ and $m=n$, then $y={{x}^{n}};\,\therefore {{D}^{n}}({{x}^{n}})=n\,!$       (iv) When $m=-1,\,\,y=\frac{1}{(ax+b)}$       ${{D}^{n}}(y)={{a}^{n}}(-1)(-2)(-3)........(-n){{(ax+b)}^{-1-n}}$       $={{a}^{n}}{{(-1)}^{n}}(1.2.3......n){{(ax+b)}^{-1-n}}=\frac{{{a}^{n}}{{(-1)}^{n}}n\,!}{{{(ax+b)}^{n+1}}}$     (III) $\frac{{{d}^{n}}}{d{{x}^{n}}}\log (ax+b)=\frac{{{(-1)}^{n-1}}(n-1)!{{a}^{n}}}{{{(ax+b)}^{n}}}$       (IV) $\frac{{{d}^{n}}}{d{{x}^{n}}}({{e}^{ax}})={{a}^{n}}{{e}^{ax}}$         (V)  $\frac{{{d}^{n}}({{a}^{x}})}{d{{x}^{n}}}={{a}^{x}}{{(\log a)}^{n}}$     (VI) (i) $\frac{{{d}^{n}}}{d{{x}^{n}}}{{e}^{ax}}\sin (bx+c)={{r}^{n}}{{e}^{ax}}\sin (bx+c+n\varphi )$       where $r=\sqrt{{{a}^{2}}+{{b}^{2}}};\,\,\varphi ={{\tan }^{-1}}\frac{b}{a}$         (ii) $\frac{{{d}^{n}}}{d{{x}^{n}}}{{e}^{ax}}\cos (bx+c)={{r}^{n}}{{e}^{ax}}\cos (bx+c+n\varphi )$

Differentiation of a Function with Respect to Another Function

In this section we will discuss derivative of a function with respect to another function. Let $u=f(x)$ and $v=g(x)$ be two functions of $x$. Then, to find the derivative of $f(x)$ w.r.t. $g(x)$ i.e., to find $\frac{du}{dv}$ we use the following formula  $\frac{du}{dv}=\frac{du/dx}{dv/dx}$.

Methods of Differentiation

(1) Differentiation of implicit functions : If $y$ is expressed entirely in terms of $x,$ then we say that $y$ is an explicit function of $x$. For example $y=\sin \,\,x,y=\text{ }{{e}^{x}},\,\,\,y={{x}^{2}}+x+1$ etc. If $y$ is related to $x$ but cannot be conveniently expressed in the form of $y=f(x)$ but can be expressed in the form $f(x,y)=0$, then we say that $y$ is an implicit function of $x$.   Working Rule 1:     (a) Differentiate each term of $f(x,y)=0$ with respect to $x$.     (b) Collect the terms containing $dy/dx$ on one side and the terms not involving $dy/dx$ on the other side.     (c) Express $dy/dx$ as a function of $x$ or $y$ or both.     In case of implicit differentiation, $dy/dx$ may contain both $x$ and $y$.     Working Rule 2:      If $f(x,\,\,y)=$ constant, then $\frac{dy}{dx}=-\frac{\left( \frac{\partial f}{\partial x} \right)}{\left( \frac{\partial f}{\partial y} \right)}$, where $f(x)$ and $\frac{\partial f}{\partial y}$ are partial differential coefficients of $f(x,\,y)$ with respect to $x$ and $y$ respectively.     (2) Logarithmic differentiation : If differentiation of an expression or an equation is done after taking log on both sides, then it is called logarithmic differentiation. This method is useful for the function having following forms     (i) $y={{[f(x)]}^{g(x)}}$     (ii) $y=\frac{{{f}_{1}}(x).{{f}_{2}}(x).........}{{{g}_{1}}(x).{{g}_{2}}(x)........}$ where ${{g}_{i}}(x)\ne 0\,$     (where $i=\text{ }1,\text{ }2,\text{ }3,.....$), ${{f}_{i}}(x)$ and ${{g}_{i}}(x)$ both are differentiable.     (i) Case I : $y=[f{{(x]}^{g(x)}}$ where $f(x)$ and $g(x)$are functions of $x$. To find the derivative of this type of functions we proceed as follows: Let $y={{[f(x)]}^{g(x)}}$. Taking logarithm of both the sides, we have $]\,a,\,b[,$$f(x)$and then we differentiate w.r.t. $x$.     (ii) Case II : $y=\frac{{{f}_{1}}(x).{{f}_{2}}(x)}{{{g}_{1}}(x).{{g}_{2}}(x)}$     Taking logarithm of both the sides, we have     $\log y=\log [{{f}_{1}}(x)]+\log [{{f}_{2}}(x)]-\log [{{g}_{1}}(x)]-\log [{{g}_{2}}(x)]$     and differentiating w.r.t. $x,$ we get     $\frac{1}{y}\frac{dy}{dx}=\frac{{{{{f}'}}_{1}}(x)}{{{f}_{1}}(x)}+\frac{{{{{f}'}}_{2}}(x)}{{{f}_{2}}(x)}-\frac{{{{{g}'}}_{1}}(x)}{{{g}_{1}}(x)}-\frac{{{{{g}'}}_{2}}(x)}{{{g}_{2}}(x)}$     (3) Differentiation of parametric functions : Sometimes $x$ and $y$ are given as functions of a single variable, e.g., $x=\phi (t),\,\,y=\psi (t)$ are two functions and t is a variable. In such a case $x$ and $y$ are called parametric functions or parametric equations and t is called the parameter. To find $\frac{dy}{dx}$ in case of parametric functions, $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$.     (4) Differentiation of infinite series : If $y$ is given in the form of infinite series of $x$ and we have to find out $]\,a,\,b[,$, then we remove one or more terms, it does not affect the series.     (i) If $y=\sqrt{f(x)+\sqrt{f(x)+\sqrt{f(x)+....\infty }}}$, then $y=\sqrt{f(x)+y}$     $\Rightarrow$${{y}^{2}}=f(x)+y$$\Rightarrow$ $2y\frac{dy}{dx}={f}'(x)+\frac{dy}{dx}$; $\therefore$$\frac{dy}{dx}=\frac{{f}'(x)}{2y-1}$.     (ii) If $y=f{{(x)}^{f(x)}}^{f{{(x)}^{f(x).....\infty }}}$ then $y=f{{(x)}^{y}}$     $\therefore$ $\log y=y\log f(x)$;  $\frac{1}{y}\frac{dy}{dx}=\frac{y.{f}'(x)}{f(x)}+\log f(x).\frac{dy}{dx}$     $\therefore$ $\frac{dy}{dx}=\frac{{{y}^{2}}{f}'(x)}{f(x)[1-y\log f(x)]}$     (iii) If $gof(x)$ then $\frac{dy}{dx}=\frac{y{f}'(x)}{2y-f(x)}$.     (5) Differentiation of composite function : Suppose a function is given in form of $fog(x)$ or $f[g(x)]$.     Working Rule:    Differentiate applying chain rule, $\frac{d}{dx}f[g(x)]=f'[g(x)].g'(x)$

Theorems for Differentiation

Let $f(x),\,g(x)$and $u(x)$be differentiable functions     (1) If at all points of a certain interval, ${f}'(x)=0,$ then the function $f(x)$ has a constant value within this interval.     (2) Chain rule     (i) Case I : If $y$ is a function of $u$ and $u$ is a function of $x,$ then derivative of $y$ with respect to $x$ is $\frac{dy}{dx}=\frac{dy}{du}\,\frac{du}{dx}$ or $y=f(u)$ $\Rightarrow \,\frac{dy}{dx}=f'(u)\frac{du}{dx}$.     (ii) Case II : If $y$ and $x$ both are expressed in terms of $t,\,\,y$ and $x$ both are differentiable with respect to $t,$ then $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$.     (3) Sum and difference rule: Using linear property $\frac{d}{dx}(f(x)\pm g(x))=\frac{d}{dx}(f(x))\pm \frac{d}{dx}(g(x))$     (4) Product rule     (i) $\frac{d}{dx}(f(x).g(x))=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)$         (ii) $c\in (a,\,b)$     (5) Scalar multiple rule : $\frac{d}{dx}(k\,f(x))=k\frac{d}{dx}f(x)$     (6) Quotient rule : $\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right)\,=\frac{g(x)\frac{d}{dx}(f(x))-f(x)\frac{d}{dx}(g(x))}{{{(g(x))}^{2}}}$     Provided $g(x)\ne 0$.

Some Standard Differentiation

(1) Differentiation of algebraic functions : $\frac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$     In particular     (i)  $\frac{d}{dx}{{[f(x)]}^{n}}=n\,{{[f(x)]}^{\,n-1}}{f}'(x)$                  (ii) $\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}$     (iii) $\frac{d}{dx}\left( \frac{1}{{{x}^{n}}} \right)=-\frac{n}{{{x}^{n+1}}}$     (2) Differentiation of trigonometric functions :     (i) $\frac{d}{dx}\sin x=\cos x$     (ii) $\frac{d}{dx}\cos x=-\sin x$     (iii) $\frac{d}{dx}\tan x={{\sec }^{2}}x$     (iv) $\frac{d}{dx}\sec x=\sec x\tan x$     (v) $\frac{d}{dx}\text{cosec}\,x=-\text{cosec}\,x\,\cot x$     (vi) $\frac{d}{dx}\cot x=-\text{cose}{{\text{c}}^{2}}x$     (3) Differentiation of logarithmic and exponential functions :        (i) $\frac{d}{dx}\log x=\frac{1}{x}$, for $x>0$                    (ii)  $\frac{d}{dx}{{e}^{x}}={{e}^{x}}$       (iii) $\frac{d}{dx}{{a}^{x}}={{a}^{x}}\log a$, for $a>0$     (iv) $\frac{d}{dx}{{\log }_{a}}x=\frac{1}{x\log a}$, for $x>0,\,\,a>0,\,\,a\ne 1$     (4) Differentiation of inverse trigonometrical functions:     (i) $\frac{d}{dx}{{\sin }^{-1}}x=\frac{1}{\sqrt{1-{{x}^{2}}}}$, for $-1<x<1$     (ii)  $\frac{d}{dx}{{\cos }^{-1}}x=\frac{-1}{\sqrt{1-{{x}^{2}}}}$, for $-1<x<1$     (iii) $\frac{d}{dx}{{\sec }^{-1}}x=\frac{1}{|x|\sqrt{{{x}^{2}}-1}}$, for $|x|>1$     (iv) $\frac{d}{dx}\text{cose}{{\text{c}}^{-1}}x=\frac{-1}{|x|\sqrt{{{x}^{2}}-1}}$, for $|x|>1$     (v) $\frac{d}{dx}{{\tan }^{-1}}x=\frac{1}{1+{{x}^{2}}}$, for $x\in R$     (vi) $f'(c)\ge 0(f'(c)<0$, for $x\in R$     (5) Differentiation of hyperbolic functions :     (i) $\frac{d}{dx}\sinh \,x=\cosh x$             (ii) $\frac{d}{dx}\cosh \,x=\sinh \,x$     (iii) $\frac{d}{dx}\tanh \,x=\sec {{\text{h}}^{2}}x$               (iv) $\frac{d}{dx}\coth \,x=-\,\text{cosec}{{\text{h}}^{2}}x$     (v) $\frac{d}{dx}\text{sech}\,x=-\text{sech}\,x\tanh x$                  (vi) $\frac{d}{dx}\text{cosech}\,x=-\text{cosech}\,x\,\,\coth x$     (vii) $\frac{d}{dx}{{\sinh }^{-1}}x=1/\sqrt{(1+{{x}^{2}})}$     (viii) $\frac{d}{dx}{{\cosh }^{-1}}x=1/\sqrt{({{x}^{2}}-1)}$     (ix) $\frac{d}{dx}{{\tanh }^{-1}}x=1/({{x}^{2}}-1)$     (x) $\frac{d}{dx}{{\coth }^{-1}}x=1/(1-{{x}^{2}})$     (xi) $\frac{d}{dx}\sec {{\text{h}}^{-1}}x=-1/x\sqrt{(1-{{x}^{2}}})$     (xii) $\frac{d}{dx}\text{cosec}{{\text{h}}^{-1}}x=-1/x\sqrt{(1+{{x}^{2}})}$     (6) Suitable substitutions
Function Substitution Function Substitution
$\sqrt{{{a}^{2}}-{{x}^{2}}}$ $x=a\sin \theta$ or $a\cos \theta$ $\sqrt{{{x}^{2}}+{{a}^{2}}}$ $x=a\tan \theta$ or $a\cot \theta$
$\sqrt{{{x}^{2}}-{{a}^{2}}}$ $x=a\sec \theta$ or $a\cos ec\theta$ more...

Introduction

The rate of change of one quantity with respect to some another quantity has a great importance.     The rate of change of a quantity $'y'$ with respect to another quantity $'x'$ is called the derivative or differential coefficient of $y$ with respect to $x$.

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