Lecture 13: Differentation Rules


The chain rule

f(x)=xnf(x) = x^n f1(x)=n(xn1)(x1)f^1 (x) = n(x^{n-1}) * ( x^1 ) 


The Product Rule

f(x)=xzf(x) = xz  f1(x)=x1z+z1x f^1 (x) = x^1z + z^1x


The Quotient Rule

f(x)=zxf(x) = \frac{z}{x} f1(x)=z1xzx1x2 f^1(x) = \frac{z^1x - zx^1}{x^2}


The Derivative of exponentials/logarithmic functions


f(x)=logaxf(x) = log_ax  f1(x)=1xIna(x1)f^1 (x) = \frac{1}{xIna} * (x^1)

f(x)=exf(x) = e^x  f1(x)=exf^1(x) = e^x

f(x)=exf1(x)=[ex][x1]f(x) = e^xf^1(x) = [e^x] * [x^1] 

f(x)=In(x)f1(x)=1xf(x) = In(x) f1 (x) = \frac{1}{x} → if y=Inx,y1=1x×dxdx=1x×x1y = In x , y1 = \frac{1}{x} × \frac{dx}{dx} = \frac{1}{x} × x^1 

f(x)=bxf1(x)=bxIn(b)f(x) = b^x f^1(x) = b^x In(b) 

f(x)=axf1(x)=axInax1f(x) = a^x f1(x) = a^x * Ina * x^1 


Trigonometric Derivative

ddxsinx=[cosx]x1\frac{d}{dx} sin x = [cosx] * x^1

ddxcosx=[sinx]x1\frac{d}{dx} cos x = [- sin x] * x^1 

ddxtanx=[sec2x]x1\frac{d}{dx} tan x = [sec2x] * x^1 


ddxcscx=csccotx\frac{d}{dx} csc x = -csc * cot x 

ddxsecx=sectanx\frac{d}{dx} sec x = sec * tan x 

ddxcotx=csc2x\frac{d}{dx} cot x = -csc2x


ddxarcsinx=11x2\frac{d}{dx} arc sin x = \frac{1}{√1-x^2} 

ddxarccosx=11x2\frac{d}{dx} arc cos x = \frac{-1}{√ 1-x^2} 

ddxarctanx=11+x2\frac{d}{dx} arctan x = \frac{1}{1+ x^2} 

ddxarccscx=111x21\frac{d}{dx} arccsc x = \frac{-1}{1 * 1 * √ x2 - 1}

ddxarcsecx=111x21\frac{d}{dx} arcsec x = \frac{1}{1 * 1 * √x2 - 1}

ddxarccotx=11+x2\frac{d}{dx} arccot x = \frac{-1}{1 + x^2} 


Basic Rule

ddx(c)=0\frac{d}{dx} (c) = 0  , ddx=1\frac{d}{dx} = 1 

ddx(f±g)=dfdx±dgdx\frac{d}{dx} (f±g) =\frac{df}{dx} ± \frac{dg}{dx}

ddx(fg)=fdgdx+gdtdx\frac{d}{dx} (fg) = f \frac{dg}{dx} + g \frac{dt}{dx}














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