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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
$\frac {d}{dx} [2 sinx ] = 2 \frac {d}{dx} sin x =2 cos (x)$
Addition and Subtraction
$\frac {d}{dx} [f(x)+g(x)]=\frac {d}{dx} f(x) + \frac {d}{dx} g(x)$
$\frac {d}{dx} [f(x)-g(x)]=\frac {d}{dx} f(x) – \frac {d}{dx} g(x)$
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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