Komen:
jika [tex]y=ax^n[/tex] , maka [tex]\frac{dy}{dx} =nax^{n-1}[/tex]
jika [tex]y=u(x)v(x)[/tex] , maka [tex]\frac{dy}{dx} =vu' + uv'[/tex]
jika [tex]y=\frac{u(x)}{v(x)}[/tex] , maka [tex]\frac{dy}{dx} =\frac{vu'-uv'}{v^2}[/tex]
jika [tex]y=(g(x))^n[/tex] , maka [tex]\frac{dy}{dx} =n(g(x))^{n-1}\cdot g'(x)[/tex]
jika [tex]y=\ln(g(x))[/tex] , maka [tex]\frac{dy}{dx} =\frac{g'(x)}{g(x)}[/tex]
jika [tex]y=e^{g(x)}[/tex] , maka [tex]\frac{dy}{dx} =g'(x)e^{g(x)[/tex]
[tex]dy/dx=\frac{1}{dx/dy}[/tex]
Bagian a
[tex]y=\sqrt[4]{x^3} =x^{3/4}[/tex]
[tex]\frac{dy}{dx} =\frac{3}{4} x^{-1/4}=\frac{3}{4\sqrt[4]{x} }[/tex]
Bagian b
[tex]y=\frac{3x^2-4x}{2x-5}[/tex]
misalkan
[tex]u(x)=3x^2-4x \Rightarrow u'(x)=6x-4[/tex]
[tex]v(x)=2x-5 \Rightarrow v'(x)=2[/tex]
maka
[tex]\frac{dy}{dx} =\frac{vu'-uv'}{v^2}[/tex]
[tex]\frac{dy}{dx} =\frac{(2x-5)(6x-4)-(3x^2-4x)(2)}{(2x-5)^2}[/tex]
[tex]\frac{dy}{dx} =\frac{(12x^2-38x+20)-(6x^2-8x)}{(2x-5)^2}[/tex]
[tex]\frac{dy}{dx} =\frac{6x^2-30x+20}{(2x-5)^2}[/tex] untuk [tex]x\neq \frac{5}{2}[/tex]
Bagian c
[tex]y=(9+6x)^{3/2}[/tex]
[tex]\frac{dy}{dx} =\frac{3}{2} (9+6x)^{1/2}\cdot(6)=9\sqrt{9+6x}[/tex]
Bagian d
[tex]x=4-y^2[/tex]
maka
[tex]dx=-2y\ dy[/tex]
[tex]\frac{dy}{dx} =-\frac{1}{2y}[/tex]
Bagian e
[tex]x=2y^2-4[/tex]
[tex]dx=4y\ dy[/tex]
[tex]\frac{dy}{dx} =\frac{1}{4y}[/tex]
Bagian f
[tex]y=\log(x^2+1)=\frac{\ln(x^2+1)}{\ln(10)}[/tex]
[tex]\frac{dy}{dx} =\frac{1}{\ln(10)} \frac{1}{x^2+1} (2x)[/tex]
[tex]\frac{dy}{dx} = \frac{2x}{(x^2+1)\ln(10)}[/tex]
Bagian g
[tex]y=\ln(2x^3+3x)[/tex]
[tex]\frac{dy}{dx} =\frac{6x^2+3}{2x^3+3x}[/tex]
Bagian h
[tex]y=a^{4x}=e^{\ln(a)\cdot4x[/tex]
[tex]\frac{dy}{dx} =4\ln(a)\cdot e^{\ln(a)\cdot4x}=4a^{4x}\ln(a)[/tex]
Bagian i
[tex]y=e^x[/tex]
[tex]\frac{dy}{dx} =e^x[/tex]
Bagian j
[tex]y=e^{x^2+2x}[/tex]
[tex]\frac{dy}{dx} =(2x+2)e^{x^2+2x}[/tex]
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