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d) y' =\(\dfrac{\left(x^2+7x+3\right)'\left(x^2-3x\right)-\left(x^2+7x+3\right)\left(x^2-3x\right)'}{\left(x^2-3x\right)^2}\)=\(\dfrac{\left(2x+7\right)\left(x^2-3x\right)-\left(x^2+7x+3\right)\left(2x-3\right)}{\left(x^2-3x\right)^2}\)=\(\dfrac{-2x^2-6x+9}{\left(x^2-3x\right)^2}\)
b) \(3\left(1-2x\right)^{20}\left(3x-2\right)^{10}\left(-14\left(3x-2\right)+11\left(1-2x\right)\right)\)
a. \(y'=\dfrac{-1}{\left(x-1\right)}\)
b. \(y'=\dfrac{5}{\left(1-3x\right)^2}\)
c. \(y=\dfrac{\left(x+1\right)^2+1}{x+1}=x+1+\dfrac{1}{x+1}\Rightarrow y'=1-\dfrac{1}{\left(x+1\right)^2}=\dfrac{x^2+2x}{\left(x+1\right)^2}\)
d. \(y'=\dfrac{4x\left(x^2-2x-3\right)-2x^2\left(2x-2\right)}{\left(x^2-2x-3\right)^2}=\dfrac{-4x^2-12x}{\left(x^2-2x-3\right)^2}\)
e. \(y'=1+\dfrac{2}{\left(x-1\right)^2}=\dfrac{x^2-2x+3}{\left(x-1\right)^2}\)
g. \(y'=\dfrac{\left(4x-4\right)\left(2x+1\right)-2\left(2x^2-4x+5\right)}{\left(2x+1\right)^2}=\dfrac{4x^2+4x-14}{\left(2x+1\right)^2}\)
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a. \(y'=4\left(x^2+x+1\right)^3.\left(x^2+x+1\right)'=4\left(x^2+x+1\right)^3\left(2x+1\right)\)
b. \(y'=5\left(1-2x^2\right)^4.\left(1-2x^2\right)'=-20x\left(1-2x^2\right)^4\)
c. \(y'=3\left(\dfrac{2x+1}{x-1}\right)^2.\left(\dfrac{2x+1}{x-1}\right)'=3\left(\dfrac{2x+1}{x-1}\right)^2.\left(\dfrac{-3}{\left(x-1\right)^2}\right)=\dfrac{-9\left(2x+1\right)^2}{\left(x-1\right)^4}\)
d. \(y'=\dfrac{2\left(x+1\right)\left(x-1\right)^3-3\left(x-1\right)^2\left(x+1\right)^2}{\left(x-1\right)^6}=\dfrac{-x^2-6x-5}{\left(x-1\right)^4}\)
e. \(y'=-\dfrac{\left[\left(x^2-2x+5\right)^2\right]'}{\left(x^2-2x+5\right)^4}=-\dfrac{2\left(x^2-2x+5\right)\left(2x-2\right)}{\left(x^2-2x+5\right)^4}=-\dfrac{4\left(x-1\right)}{\left(x^2-2x+5\right)^3}\)
f. \(y'=4\left(3-2x^2\right)^3.\left(3-2x^2\right)'=-16x\left(3-2x^2\right)^3\)
a: Ta có: \(y=\left(x+1\right)\left(\sqrt{x}-1\right)\)
=>\(y^{\prime}=\left(x+1\right)^{\prime}\cdot\left(\sqrt{x}-1\right)+\left(x+1\right)\cdot\left(\sqrt{x}-1\right)^{\prime}=\left(\sqrt{x}-1\right)+\left(x+1\right)\cdot\left(\frac{1}{2\sqrt{x}}\right)=\frac{x+1}{2\sqrt{x}}+\sqrt{x}-1\)
\(=\frac{x+1+2x-2\sqrt{x}}{2\sqrt{x}}=\frac{3x-2\sqrt{x}+1}{2\sqrt{x}}\)
b: ta có: \(y=\left(x^2-3\right)\left(x^3+3x^2-5\right)\)
=>\(y^{\prime}=\left(x^2-3\right)^{\prime}\left(x^3+3x^2-5\right)+\left(x^2-3\right)\left(x^3+3x^2-5\right)^{\prime}\)
=>\(y^{\prime}=2x\left(x^3+3x^2-5\right)+\left(x^2-3\right)\left(3x^2+6x\right)\)
\(=2x^4+6x^3-10x+3x^4+6x^3-9x^2-18x=5x^4+12x^3-9x^2-28x\)