# Laws of Differentiation

### Algebraic functions

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

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

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

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

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

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

$(7) \,\,\,\,\,$ $\dfrac{d}{dx} \, |x| = \dfrac{|x|}{x}$

$(8) \,\,\,\,\,$ $\dfrac{d}{dx} \, \log |x| = \dfrac{1}{x}$

$(9) \,\,\,\,\,$ $\dfrac{d}{dx} \, x^x = x^x (1+\log x)$

### 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) $-\operatorname{cosec^2}x$

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

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

### Inverse Trigonometric functions

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

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

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

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

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

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

### Hyperbolic functions

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

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

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

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

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

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

### Inverse Hyperbolic functions

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

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

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

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

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

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

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