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.

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All Chapter Notes, Concept and Important Formula

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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 $\dfrac{d y}{d x}=f^{\prime}(u) \cdot \dfrac{d u}{d x}$.

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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}$

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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.

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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.

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6. PARAMETRIC DIFFERENTIATION :

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

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

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

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8. DERIVATIVE OF A FUNCTION AND ITS INVERSE FUNCTION :

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

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

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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.

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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.

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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 $[f(x)]^{\phi(x)}=e^{\phi(x) \cdot\ln f(x)}$

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