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