Methods-of-differentiation Formula Sheet

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.

Neetesh Kumar | May 29, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Derivative by First Principle
• 2. Fundamental Rules of Differentiation
• 3. Derivative of Standard Functions
• 4. Logarithmic Differentiation
• 5. Differentiation of Implicit Function
• 6. Parametric Differentiation
• 7. Derivative of a Function w.r.t. another Function
• 8. Derivative of Inverse Function
• 9. Higher Order derivatives
• 10. Differentiation of Determinants
• 11. L Hopital's Rule

1. Derivative by First Principle:

Obtaining the derivative using the definition
$\lim\limits_{\delta x\to0} \frac{\delta y}{\delta x} = \lim\limits_{\delta x\to0} \frac{f(x+\delta x ) - f(x)}{\delta x} = f'(x) = \frac{dy}{dx}$
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
$\bold{(a)}$ $\frac{d}{dx}(f \plusmn g) = \frac{df}{dx} \plusmn \frac{dy}{dx}$ is known as $\bold{Sum \space Rule}$

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

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

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

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

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

3. Derivative of Standard Functions:

$f(x)$	$f'(x)$
$x^n$	$nx^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a, a > 0$
$\ln x$	$\frac{1}{x}$
$\log_a x$	$\frac{1}{x \ln a}, a > 0, a \neq 1$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$
$\cot x$	$-\csc^2 x$
constant	0
$\sin^{-1} x$	$\frac{1}{\sqrt{1-x^2}}, -1 < x < 1$
$\cos^{-1} x$	$-\frac{1}{\sqrt{1-x^2}}, -1 < x < 1$
$\tan^{-1} x$	$\frac{1}{1+x^2}, x \in \mathbb{R}$

| $\sec^{-1} x$ | $\frac{1}{|x| \sqrt{x^2-1}}$, |x| > 1 |

| $\csc^{-1} x$ | $-\frac{1}{|x| \sqrt{x^2-1}}, |x| > 1$ |

| $\cot^{-1} x$ | $-\frac{1}{1+x^2}, x \in \mathbb{R}$ |

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 $\frac{dy}{dx}$, 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 $\frac{dy}{dx}$ together on one side to finally find $\frac{dy}{dx}$ OR $\frac{dy}{dx} = -\frac{\partial \phi / \partial x}{\partial \phi / \partial y}$

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

(b) In the expression of $\frac{dy}{dx}$ 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
$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$.

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

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

8. Derivative of Inverse Function:

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

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 $\frac{dy}{dx}$ 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 $\frac{d^2y}{dx^2}$ or $y''$. Similarly, the 3rd order derivative of $y$ w.r.t. $x$, if it exists, is defined by $\frac{d^3y}{dx^3} = \frac{d}{dx} \left( \frac{d^2y}{dx^2} \right)$. 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 $\frac{0}{0}, \frac{\infty}{\infty}$. If the functions $f(x)$ and $g(x)$ are differentiable and $g'(x) \neq 0$ near $a$ (except possibly at $a$),
then: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ provided the limit on the right side exists.

(b) If the result is again of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, we can apply L'Hopital's Rule again: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)}$ 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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