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Math formulas: Common derivatives

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Basic Properties of Derivatives

(cf(x))=cf(x) \left(c \cdot f(x)\right)' = c \cdot f'(x)
(f±g)=f±g \left(f \pm g \right)' = f' \pm g'

Product rule

(fg)=fg+fg (f \cdot g)' = f' \cdot g + f \cdot g'

Quotient rule

(fg)=fgfgg2 \left( \frac{f}{g} \right)' = \frac{ f'\cdot g - f \cdot g' }{g^2}

Chain rule

(f(g(x)))=f(g(x))g(x) \left( f \left(g(x) \right) \right)' = f'(g(x)) \cdot g'(x)

Common Derivatives

ddx(C)=0 \frac{d}{dx} (C) = 0
ddx(x)=0 \frac{d}{dx} (x) = 0
ddx(xn)=nxn1 \frac{d}{dx} (x^n) = n \cdot x^{n-1}
ddx(sinx)=cosx \frac{d}{dx} (\sin x) = \cos x
ddx(cosx)=sinx \frac{d}{dx} (\cos x) = -\sin x
ddx(tanx)=1cos2x \frac{d}{dx} (\tan x) = \frac{1}{\cos^2x}
ddx(secx)=secxtanx \frac{d}{dx} ( \sec x) = \sec x \cdot \tan x
ddx(cscx)=cscxcotx \frac{d}{dx} (\csc x) = - \csc x \cdot \cot x
ddx(cotx)=1sin2x \frac{d}{dx} (\cot x) = -\frac{1}{ \sin^2x }
ddx(arcsinx)=11x2 \frac{d}{dx} (\arcsin x) = \frac{1}{ \sqrt{1-x^2} }
ddx(arccosx)=11x2 \frac{d}{dx} (\arccos x) = -\frac{1}{\sqrt{1-x^2}}
ddx(arctanx)=11+x2 \frac{d}{dx} (\arctan x) = \frac{1}{1+x^2}
ddx(ax)=axlna \frac{d}{dx} (a^x) = a^x \cdot \ln a
ddx(ex)=ex \frac{d}{dx} (e^x) = e^x
ddx(lnx)=1x,x>0 \frac{d}{dx} (\ln x) = \frac{1}{x} , x > 0
ddx(lnx)=1x,x0 \frac{d}{dx} (\ln |x|) = \frac{1}{x} , x \ne 0
ddx(logax)=1xlna,x>0 \frac{d}{dx} \left( \log_a x \right) = \frac{1}{x\cdot \ln a} , x > 0

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