Method of differntiation - Notes, Concept and All Important Formula

METHODS OF DIFFERENTIATION

1. DERIVATIVE OF f(x) FROM THE FIRST PRINCIPLE :

Obtaining the derivative using the definition $\displaystyle\displaystyle \lim_{\delta x \rightarrow 0} \dfrac{\delta y}{\delta x}= \displaystyle\displaystyle \lim_{\delta x \rightarrow 0} \dfrac{f(x+\delta x)-f(x)}{\delta x}=f^{\prime}(x)=\dfrac{d y}{d x}$ is called calculating derivative using first principle or ab initio or delta method.

All Chapter Notes, Concept and Important Formula

2. FUNDAMENTAL THEOREMS :

If $f$ and $g$ are derivable function of $x$, then,

(a) $\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{f} \pm \mathrm{g})=\dfrac{\mathrm{df}}{\mathrm{dx}} \pm \dfrac{\mathrm{d} \mathrm{g}}{\mathrm{d} \mathrm{x}}$, known as SUM RULE

(b) $\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{cf})=\mathrm{c} \dfrac{\mathrm{df}}{\mathrm{dx}}$, where $\mathrm{c}$ is any constant

(c) $\dfrac{\mathrm{d}}{\mathrm{dx}}(\mathrm{fg})=\mathrm{f} \dfrac{\mathrm{dg}}{\mathrm{dx}}+\mathrm{g} \dfrac{\mathrm{df}}{\mathrm{dx}}$, known as PRODUCT RULE

(d) $\dfrac{\mathrm{d}}{\mathrm{dx}}\left(\dfrac{\mathrm{f}}{\mathrm{g}}\right)=\dfrac{\mathrm{g}\left(\dfrac{\mathrm{df}}{\mathrm{dx}}\right)-\mathrm{f}\left(\dfrac{\mathrm{d} g}{\mathrm{dx}}\right)}{\mathrm{g}^{2}}$

where $g \neq 0$ known as QUOTIENT RULE

(e) If $y=f(u) $ & $u=g(x)$, then $\dfrac{d y}{d x}=\dfrac{d y}{d u} \cdot \dfrac{d u}{d x}$, known as CHAIN RULE

Note : In general if $y=f(u)$, then $\dfrac{d y}{d x}=f^{\prime}(u) \cdot \dfrac{d u}{d x}$.

3. DERIVATIVE OF STANDARD FUNCTIONS :

$\begin{array}{l} \begin{array}{|c|c|c|} \hline S.No & f(x) & f'(x) \\ \hline \text{(i)} & x^n & nx^{n-1} \\ \text{(ii)} & e^x & e^x \\ \text{(iii)} & a^x & a^x\, \ln a \, , a\gt 0 \\ \text{(iv)} & \ln x & 1/x \\ \text{(v)} & \log_a x & (1/x)\log_a e \, , a\gt 0 \, a\ne 1 \\ \text{(vi)} & \sin x & \cos x \\ \text{(vii)} & \cos x & -\sin x \\ \text{(viii)} & \tan x & \sec^2 x \\ \text{(xi)} & \sec x & \sec x \tan x \\ \text{(x)} & \text{cosec x} & -\text{cosec x}\cot x \\ \text{(xi)} & \cot x & -\text{cosec$^2$ }x \\ \text{(xii)} & \text{constant} & 0 \\ \text{(xiii)} & \sin^{-1}x & \dfrac{1}{\sqrt{1-x^2}} \,\, -1\lt x\lt 1 \\ \text{(xiv)} & \cos^{-1}x & \dfrac{-1}{\sqrt{1-x^2}} \,\, -1\lt x\lt 1\\ \text{(xv)} & \tan^{-1}x & \dfrac{1}{1+x^2} \, \, x \in R\\ \text{(xvi)} & \sec^{-1}x & \dfrac{1}{|x|\sqrt{x^2-1}} \,\, |x|\gt 1 \\ \text{(xvii)} & \text{cosec$^{-1}$}x & \dfrac{-1}{|x|\sqrt{x^2-1}}\,\, |x|\gt 1 \\ \text{(xviii)} & \cot^{-1}x & \dfrac{-1}{1+x^2}\, \, x \in R \\ \hline \end{array} \end{array}$

4. LOGARITHMIC DIFFERENTIATION :

To find the derivative of :

(a) A function which is the product or quotient of a number of function or

(b) A function of the form $[f(x)]^{g (x)}$ where f & g are both derivable, it is convenient to take the logarithm of the function first & then differentiate.

5. DIFFERENTIATION OF IMPLICIT FUNCTION:

(a) Let function is $\phi(x, y)=0$ then to find $d y / d x$, in the case of implicit functions, we differentiate each term w.r.t. x regarding y as a functions of $x$ & then collect terms in dy / dx together on one side to finally find $\mathrm{dy} / \mathrm{dx}.$

OR $\dfrac{\mathrm{dy}}{\mathrm{dx}}=\dfrac{-\partial \phi / \partial \mathrm{x}}{\partial \phi / \partial \mathrm{y}}$ where $\dfrac{\partial \phi}{\partial \mathrm{x}} $ & $\dfrac{\partial \phi}{\partial \mathrm{y}}$ are partial differential coefficient of $\phi(x, y)$ w.r.t $x $ & $y$ respectively.

(b) In expression of $\mathrm{dy} / \mathrm{dx}$ in the case of implicit functions, both x & y are present.

6. PARAMETRIC DIFFERENTIATION :

If $y=f(\theta) $ & $x=g(\theta)$ where $\theta$ is a parameter, then $\dfrac{d y}{d x}=\dfrac{d y / d \theta}{d x / d \theta}$.

7. DERIVATIVE OF A FUNCTIONW.R.T. ANOTHER FUNCTION:

Let $y=f(x) ; \quad z=g(x)$, then $\dfrac{d y}{d z}=\dfrac{d y / d x}{d z / d x}=\dfrac{f^{\prime}(x)}{g^{\prime}(x)}$

8. DERIVATIVE OF A FUNCTION AND ITS INVERSE FUNCTION :

If inverse of $y=f(x)$ is denoted as $g(x)=f^{-1}(x)$, then $g(f(x))=x$

$\Rightarrow g^{\prime}(f(x)) f^{\prime}(x)=1$

9. HIGHER ORDER DERIVATIVE :

Let a function $y=f(x)$ be defined on an open interval $(a, b)$. It's derivative, if it exists on $(\mathrm{a}, \mathrm{b})$ is a certain function $\mathrm{f}^{\prime}(\mathrm{x})$ [or $(\mathrm{dy} / \mathrm{dx})$ or $\left.y^{\prime}\right] \&$ it is called the first derivative of $y$ w. r. t. $x$. If it happens that the first derivative has a derivative on $(a, b)$ then this derivative is called second derivative of y w.r.t. x & is denoted by $f^{\prime \prime}(x)$ or $\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)$ or $\mathrm{y}^{\prime \prime}$. Similarly, the $3^{\text {rd }}$ order derivative of $y$ w.r.t $\mathrm{x}$, if it exists, is defined by $\dfrac{d^{3} y}{d x^{3}}=\dfrac{d}{d x}\left(\dfrac{d^{2} y}{d x^{2}}\right) .$ It is also denoted by $f^{\prime \prime \prime}(x)$ or $y^{\prime \prime \prime} $ & so on.

10. DIFFERENTIATION OF DETERMINANTS:

If $F(x)=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l(x) & m(x) & n(x) \\ u(x) & v(x) & w(x)\end{array}\right|$, where $f, g$, h. $l, m, n, u, v, w$ are differentiable functions of $\mathrm{x}$, then

$F^{\prime}(x)=\left|\begin{array}{lll}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l(x) & m(x) & n(x) \\ u(x) & v(x) & w(x)\end{array}\right|$$+\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l^{\prime}(x) & m^{\prime}(x) & n^{\prime}(x) \\ u(x) & v(x) & w(x)\end{array}\right|$$+\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l(x) & m(x) & n(x) \\ u^{\prime}(x) & v^{\prime}(x) & w^{\prime}(x)\end{array}\right|$

Similarly one can also proceed column wise.

11. L' HÔPITAL'S RULE :

(a) Applicable while calculating limits of indeterminate forms of the type $\dfrac{0}{0}, \dfrac{\infty}{\infty}$. If the function $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are differentiable in certain neighbourhood of the point $\mathrm{a}$, except, may be, at the point a itself, and $g^{\prime}(x) \neq 0$, and if

$\displaystyle \lim _{x \rightarrow a} f(x)=\displaystyle \lim _{x \rightarrow a} g(x)=0$ or $\displaystyle \lim _{x \rightarrow a} f(x)=\displaystyle \lim _{x \rightarrow a} g(x)=\infty$,

then $\displaystyle \lim _{x \rightarrow a} \dfrac{f(x)}{g(x)}=\displaystyle \lim _{x \rightarrow a} \dfrac{f^{\prime}(x)}{g^{\prime}(x)}$

provided the limit $\displaystyle \lim _{x \rightarrow a} \dfrac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{g}^{\prime}(\mathrm{x})}$ exists (L' Hôpital's rule). The point 'a' may be either finite or improper $+\infty$ or $-\infty$.

(b) Indeterminate forms of the type $0 . \infty$ or $\infty-\infty$ are reduced to forms of the type $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ by algebraic transformations.

(c) Indeterminate forms of the type $1^{\infty}, \infty^{0}$ or $0^{0}$ are reduced to forms of the type $0 . \infty$ by taking logarithms or by the transformation $[f(x)]^{\phi(x)}=e^{\phi(x) \cdot\ln f(x)}$

Solultion of Triangle - Notes, Concept and All Important Formula

PROPERTIES AND SOLUTIONS OF TRIANGLE 1. SINE FORMULAE : In any triangle $\mathrm{ABC}$ $\dfrac{\mathrm{a}}{\sin \mathrm{A}}=\dfrac{\mathrm{b}}{\sin \mathrm{B}}=\dfrac{\mathrm{c}}{\sin \mathrm{C}}=\lambda=\dfrac{\mathrm{abc}}{2 \Delta}=2 \mathrm{R}$ where $\mathrm{R}$ is circumradius and $\Delta$ is area of triangle. All Chapter Notes, Concept and Important Formula 2. COSINE FORMULAE : (a) $\mathrm{\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2 b c}}$ or $\mathrm{a^{2}=b^{2}+c^{2}-2 b c \cos A}$ (b) $\mathrm{\cos B=\dfrac{c^{2}+a^{2}-b^{2}}{2 c a}}$ (c) $\mathrm{\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2 a b}}$ 3. PROJECTION FORMULAE : (a) $b \cos C+c \cos B=a$ (b) $c \cos A+a \cos C=b$ (c) $a \cos B+b \cos A=c$ 4. NAPIER'S ANALOGY (TANGENT RULE) : (a) $\tan \left(\dfrac{\mathrm{B}-\mathrm{C}}{2}\right)=\dfrac{\mathrm{b}-\mathrm{c}}{\mathrm{b}+\mathrm{c}} \cot \dfrac{\mathrm{A}}{2}$ (b) \(\tan \left(\dfrac{\mathrm{C}-\mathrm{A}}{2}\right)=\dfrac{\mathrm{c}-\mathrm{...

MATHEMATICAL WORLD

Search This Blog