Math Doubts

Derivative Rules

Properties

$(1) \,\,\,\,\,$ $\dfrac{d}{dx}{\, \Big(f{(x)}+g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $+$ $\dfrac{d}{dx}{\, g{(x)}}$

$(2) \,\,\,\,\,$ $\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $-$ $\dfrac{d}{dx}{\, g{(x)}}$

$(3) \,\,\,\,\,$ $\dfrac{d}{dx}{\, \Big(f{(x)}.g{(x)}\Big)}$ $\,=\,$ ${f{(x)}}{\dfrac{d}{dx}{\, g{(x)}}}$ $+$ ${g{(x)}}{\dfrac{d}{dx}{\, f{(x)}}}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx}{\, \Bigg(\dfrac{f{(x)}}{g{(x)}}\Bigg)}$ $\,=\,$ $\dfrac{{g{(x)}}{\dfrac{d}{dx}{f{(x)}}}-{f{(x)}}{\dfrac{d}{dx}{g{(x)}}}}{{g{(x)}}^2}$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx}\,f\Big(g(x)\Big)$ $\,=\,$ $f’\Big(g(x)\Big).g'(x)$

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Formulas

The list of standard results to calculating derivative of standard functions.

Algebraic functions

$(1) \,\,\,\,\,$ $\dfrac{d}{dx} \, c = 0$

$(2) \,\,\,\,\,$ $\dfrac{d}{dx} \, x = 1$

$(3) \,\,\,\,\,$ $\dfrac{d}{dx} \, x^n = nx^{n-1}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx} \, e^x = e^x$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx} \, a^x = a^x \log a$

$(6) \,\,\,\,\,$ $\dfrac{d}{dx} \, \log_{e}{x} = \dfrac{1}{x}$

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Trigonometric functions

$(1) \,\,\,\,\,$ $\dfrac{d}{dx} \, \sin{x} = \cos{x}$

$(2) \,\,\,\,\,$ $\dfrac{d}{dx} \, \cos{x} = -\sin{x}$

$(3) \,\,\,\,\,$ $\dfrac{d}{dx} \, \tan{x} = \sec^2{x}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx} \, \cot{x} = -\csc^2{x}$ (or) $-\operatorname{cosec^2}x$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx} \, \sec{x} = \sec{x}\cot{x}$

$(6) \,\,\,\,\,$ $\dfrac{d}{dx} \, \csc{x} = -\csc{x}\cot{x}$

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Inverse Trigonometric functions

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

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

$(3) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\tan^{-1}{x}} \,=\, \dfrac{1}{1+x^2}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\cot^{-1}{x}} \,=\, \dfrac{-1}{1+x^2}$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\sec^{-1}{x}} \,=\, \dfrac{1}{|x|\sqrt{x^2-1}}$

$(6) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\csc^{-1}{x}} \,=\, \dfrac{-1}{|x|\sqrt{x^2-1}}$

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Hyperbolic functions

$(1) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\sinh{x}} \,=\, \cosh{x}$

$(2) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\cosh{x}} \,=\, \sinh{x}$

$(3) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\tanh{x}} \,=\, \operatorname{sech}^2{x}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\coth{x}} \,=\, -\operatorname{csch}^2{x}$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\operatorname{sech}{x}} \,=\, -\operatorname{sech}{x}\tanh{x}$

$(6) \,\,\,\,\,$ $\dfrac{d}{dx}{\,\operatorname{csch}{x}} \,=\, -\operatorname{csch}{x}\coth{x}$

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Inverse Hyperbolic functions

$(1) \,\,\,\,\,$ $\dfrac{d}{dx} \, \sinh^{-1} x = \dfrac{1}{\sqrt{1+x^2}}$

$(2) \,\,\,\,\,$ $\dfrac{d}{dx} \, \cosh^{-1} x = \dfrac{1}{\sqrt{x^2 -1}}$

$(3) \,\,\,\,\,$ $\dfrac{d}{dx} \, \tanh^{-1} x = \dfrac{1}{1-x^2}$

$(4) \,\,\,\,\,$ $\dfrac{d}{dx} \, \coth^{-1} x = \dfrac{1}{1-x^2}$

$(5) \,\,\,\,\,$ $\dfrac{d}{dx} \, \operatorname{sech}^{-1} x = \dfrac{-1}{|x| \sqrt{1 -x^2}}$

$(6) \,\,\,\,\,$ $\dfrac{d}{dx} \, \operatorname{csch}^{-1} x = \dfrac{-1}{|x| \sqrt{x^2 +1}}$