# Differentiation formulas |Derivatives of Function list

Here is the list of Differentiation formulas|Derivatives of Function to remember to score well in your Mathematics examination. The formula list include the derivative of polynomial functions, trigonometric functions,inverse trigonometric function, Logarithm function,exponential function. This also includes the rules for finding the derivative of various composite function and difficult function,

## Differentiation from First Principle

$\frac {d}{dx} f(x) =\lim_{h\rightarrow 0} \frac{f(x+h) -f(x)}{h}$

## Standard Differentiation formulas

$\frac {d}{dx} (c) = 0$  ( Where c is a constant)

$\frac {d}{dx} (cx) = c$ ( Where c is a constant)

$\frac {d}{dx} (x^n) = nx^{n-1}$

$\frac {d}{dx} (e^x) = e^x$

$\frac {d}{dx} (ln x) = \frac {1}{x}$

$\frac {d}{dx} (log_{10} x) =\frac {1}{x ln 10}$

$\frac {d}{dx} (log_{a} x) =\frac {1}{x ln a}$

$\frac {d}{dx} (a^x) = a^x ln a$

## Differentiation formulas for Trigonometric Functions

$\frac {d}{dx} (sinx) = cos x$

$\frac {d}{dx} (cos x) = -sin x$

$\frac {d}{dx} (tan x) = sec^2 x, x \ne (2n +1) \frac {\pi}{2}, n \in I$

$\frac {d}{dx} (cot x) = -cosec^2 x, x \ne n \pi, n \in I$

$\frac {d}{dx} (sec x) = sec(x) tan(x) ,x \ne (2n +1) \frac {\pi}{2}, n \in I$

$\frac {d}{dx} (cosec x) = -cosec(x) cot(x) , x \ne n \pi, n \in I$

## Differentiation formulas for Inverse Trigonometric Functions

$\frac {d}{dx} (sin^{-1}x) = \frac {1}{\sqrt {1-x^2} }, -1< x< 1$

$\frac {d}{dx} (cos ^{-1}x) = -\frac {1}{\sqrt {1-x^2}} , -1< x< 1$

$\frac {d}{dx} (tan ^{-1}x) = \frac {1}{1 + x^2}$

$\frac {d}{dx} (cot ^{-1}x) = -\frac {1}{1 + x^2}$

$\frac {d}{dx} (sec ^{-1}x) = \frac {1}{|x|\sqrt {x^-1}} , |x| > 1$

$\frac {d}{dx} (cosec ^{-1}x) = \frac {1}{|x|\sqrt {x^-1}} , |x| > 1$

## Algebra of derivatives

Multiplication by Constant

$\frac {d}{dx} [cf(x)] = c \frac {d}{dx} f(x)$

Example

$\frac {d}{dx} [2 sinx ] = 2 \frac {d}{dx} sin x =2 cos (x)$

$\frac {d}{dx} [f(x)+g(x)]=\frac {d}{dx} f(x) + \frac {d}{dx} g(x)$

$\frac {d}{dx} [f(x)-g(x)]=\frac {d}{dx} f(x) – \frac {d}{dx} g(x)$

Example

$\frac {d}{dx} [sinx -cos x ] = \frac {d}{dx} sin x – \frac {d}{dx} cos x =cos (x) + sin(x)$

Multiplication

$\frac {d}{dx} [f(x)g(x)]=g(x) \frac {d}{dx} f(x) + f(x) \frac {d}{dx} g(x)$

Example

$\frac {d}{dx} [x^2 sinx ] = x^2 \frac {d}{dx} sin x + sin x \frac {d}{dx} x^2 =x^2 cos (x) + 2x sin(x)$

Division

$\frac {d}{dx} [f(x)/g(x)]=\frac {g(x) \frac {d}{dx} f(x) – f(x) \frac {d}{dx} g(x)}{[g(x)]^2}$

Example

$\frac {d}{dx} [sin(x) /x^2]=\frac {x^2 \frac {d}{dx} sin(x) – sin(x) \frac {d}{dx} x^2}{x^4}$

$=\frac {x^2 cos (x) – 2x sin(x)}{x^4}$

$=\frac {x cos (x) – 2 sin(x)}{x^3}$

## Chain Rule

if y = f(u) and u =g(x) ,then

$\frac {dy}{dx} = \frac {dy}{du} \frac {du}{dx}$

Example

$\frac {d}{dx} [sin (x^3)] = \frac {d}{du} sin (u) \frac {d}{dx} (x^3)= 3x^2 cos (x^3)$

## Differentiable at a point a

$\lim_{h \rightarrow 0 -0} \frac{f(a+h) -f(a)}{h} = \lim_{h \rightarrow 0 +0} \frac{f(a+h) -f(a)}{h}= \text {finite number}$

## Logarithmic Differentiation

If  $y = \frac {f_1(x) f_2(x) f_3(x)}{ g_1(x) g_2(x) g_3(x)}$

Then we first take logarithm and then differentiate it

## nth Derivatives of Common Function

$\frac {d^n}{dx^n} [sin(ax + b)]= a^n sin (\frac {n\pi}{2} + ax + b)$

$\frac {d^n}{dx^n} [cos(ax + b)]= a^n cos (\frac {n\pi}{2} + ax + b)$

$\frac {d^n}{dx^n} [e^{ax}]= a^n e^{ax}$

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