Methods-of-differentiation Formula Sheet

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Differentiation is the process in calculus of finding the derivative of a function, which describes the rate at which the function changes concerning its independent variable.

1. Derivative by First Principle:

Obtaining the derivative using the definition
$\lim\limits_{\delta x\to0} \dfrac{\delta y}{\delta x} = \lim\limits_{\delta x\to0} \dfrac{f(x+\delta x ) - f(x)}{\delta x} = f'(x) = \dfrac{dy}{dx}$
is called calculating the derivative using the first principle or 'ab initio' or delta method.

2. Fundamental Rules of Differentiation:

If f and g are derivable functions of x, then
$\bold{(a)}$ $\dfrac{d}{dx}(f \plusmn g) = \dfrac{df}{dx} \plusmn \dfrac{dy}{dx}$ is known as $\bold{Sum \space Rule.}$

$\bold{(b)}$ $\dfrac{d}{dx}(c.f) = c.\dfrac{df}{dx},$ is known as $\bold{Constant \space Multiple \space Rule}$ where c is any constant.

$\bold{(c)}$ $\dfrac{d}{dx} (f.g) = f .\dfrac{dg}{dx} + g. \dfrac{df}{dx}$, known as $\bold{Product \space Rule.}$

$\bold{(d)}$ $\dfrac{d}{dx} \left( \dfrac{f}{g} \right) = \dfrac{g .\dfrac{df}{dx} - f .\dfrac{dg}{dx}}{g^2}$, where $g \neq 0$, known as $\bold{Quotient \space Rule.}$

$\bold{(e)}$ If $y = f(u)$ and $u = g(x)$, then $\dfrac{dy}{dx} = \dfrac{dy}{du}.\dfrac{du}{dx}$, known as $\bold{Chain \space Rule.}$

Note: In general, if $y = f(u)$, then $\dfrac{dy}{dx} = f'(u) \cdot \dfrac{du}{dx}$.

3. Derivative of Standard Functions:

$f(x)$	$f'(x)$
$Constant$	$0$
$x^n$	$n \cdot x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x .\ln a$
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x. \ln a}$
$abs(x)$	$\dfrac{x}{abs(x)} \text{ (for } x \neq 0 \text{)}$
$\sqrt{x}$	$\dfrac{1}{2.\sqrt{x}}$
$\sqrt[n]{x}$	$\dfrac{1}{n}. x^{(\frac{1}{n} - 1)}$
$\sin (x)$	$\cos (x)$
$\cos (x)$	$-\sin (x)$
$\tan (x)$	$\sec^2 (x)$
$\cot (x)$	$-\csc^2 (x)$
$\sec (x)$	$\sec (x) .\tan (x)$
$\csc (x)$	$-\csc (x) .\cot (x)$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$	$-\dfrac{1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$
$\cot^{-1} x$	$-\dfrac{1}{1+x^2}$
$\sec^{-1} x$	$\dfrac{1}{abs(x). \sqrt{x^2 - 1}}$
$\csc^{-1} x$	$-\dfrac{1}{abs(x). \sqrt{x^2 - 1}}$
$\sinh (x)$	$\cosh (x)$
$\cosh (x)$	$\sinh (x)$
$\tanh (x)$	$\text{sech}^2 (x)$
$\coth (x)$	$-\text{csch}^2 (x)$
$\text{sech} (x)$	$-\text{sech} (x). \tanh (x)$
$\text{csch} (x)$	$-\text{csch} (x). \coth (x)$
$\sinh^{-1} (x)$	$\dfrac{1}{\sqrt{x^2 + 1}}$
$\cosh^{-1} (x)$	$\dfrac{1}{\sqrt{x^2 - 1}}$
$\tanh^{-1} (x)$	$\dfrac{1}{1 - x^2}$
$\coth^{-1} (x)$	$\dfrac{1}{1 - x^2}$
$\text{sech}^{-1} (x)$	$-\dfrac{1}{x. \sqrt{1 - x^2}}$
$\text{csch}^{-1} (x)$	$-\dfrac{1}{abs(x). \sqrt{1 + x^2}}$

4. Logarithimic Differentiation:

To find the derivative of:

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

(b) A function of the form $[f(x)]^{g(x)}$ where $f$ and $g$ are both derivable,

it is convenient to take the logarithm of the function first and then differentiate.

5. Differentiation of Implicit Function:

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

$\dfrac{dy}{dx} = -\dfrac{\partial \phi / \partial x}{\partial \phi / \partial y}$

where $\dfrac{\partial \phi}{\partial x}$ and $\dfrac{\partial \phi}{\partial y}$ are partial differential coefficients of $\phi(x, y)$ w.r.t. $x$ and $y$ respectively.

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

6. Parametric Differentiation:

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

7. Derivative of a Function w.r.t. another Function:

Let $y = f(x)$ and $z = g(x)$, then $\dfrac{dy}{dz} = \dfrac{\dfrac{dy}{dx}}{\dfrac{dz}{dx}} = \dfrac{f'(x)}{g'(x)}$.

8. Derivative of Inverse Function:

If the inverse of $y = f(x)$ is denoted as $g(x) = f^{-1}(x)$, then $g\bigg(f(x)\bigg) = x$ $\Rightarrow g'\bigg(f(x)\bigg). f'(x) = 1$.

9. Higher Order derivatives:

Let a function $y = f(x)$ be defined on an open interval $(a, b)$. Its derivative, if it exists on $(a, b)$, is a certain function $f'(x)$ [or $\dfrac{dy}{dx}$ or $y'$] and it is called the first derivative of $y$ w.r.t. $x$.

If the first derivative has a derivative on $(a, b)$, then this derivative is called the second derivative of $y$ w.r.t. $x$ and is denoted by $f''(x)$ or $\dfrac{d^2y}{dx^2}$ or $y''$.

Similarly, the 3rd order derivative of $y$ w.r.t. $x$, if it exists, is defined by $\dfrac{d^3y}{dx^3} = \dfrac{d}{dx} \left( \dfrac{d^2y}{dx^2} \right)$. It is also denoted by $f'''(x)$ or $y'''$.

10. Differentiation of Determinants:

If $F(x) = \begin{vmatrix} f(x) & g(x) & h(x) \\ \lambda(x) & m(x) & n(x) \\ u(x) & v(x) & w(x) \end{vmatrix}$ where $f, g, h, \lambda, m, n, u, v, w$ are differentiable functions of $x$, then

$F'(x) = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ \lambda(x) & m(x) & n(x) \\ u(x) & v(x) & w(x) \end{vmatrix}$ + $\begin{vmatrix} f(x) & g(x) & h(x) \\ \lambda'(x) & m'(x) & n'(x) \\ u(x) & v(x) & w(x) \end{vmatrix}$ + $\begin{vmatrix} f(x) & g(x) & h(x) \\ \lambda(x) & m(x) & n(x) \\ u'(x) & v'(x) & w'(x) \end{vmatrix}$

Similarly, one can also proceed column-wise.

11. L Hopital's Rule:

(a) Applicable while calculating limits of indeterminate forms of the type $\dfrac{0}{0}, \dfrac{\infty}{\infty}$.

If the functions $f(x)$ and $g(x)$ are differentiable and $g'(x) \neq 0$ near $a$ (except possibly at $a$),
then: $\lim\limits_{x \to a} \dfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \dfrac{f'(x)}{g'(x)}$ provided the limit on the right side exists.

(b) If the result is again of the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, we can apply L'Hopital's Rule again:

$\lim\limits_{x \to a} \dfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \dfrac{f''(x)}{g''(x)}$ and so on, until the limit is not in indeterminate form.

(c) For indeterminate forms of the type $0 \cdot \infty$, $\infty - \infty$, $0^0$, $\infty^0$, and $1^\infty$, appropriate algebraic manipulations can be made to convert them into a form where L'Hopital's Rule can be applied.

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