Question

Тест з диференціювання(похідні функції), часть 3

Answer 1

$f(x)=\left(x-5\right)\left(2x-5\right)\\\\\big | \: \left(f\cdot g\right)'=f\:'\cdot g+f\cdot g'\\\\f'(x)=\left(x-5\right)'\left(2x-5\right)+\left(2x-5\right)'\left(x-5\right)=\\= 1\cdot \left(2x-5\right)+2\left(x-5\right)=\\=2x-5+2x-10=\\=4x-15$

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$f(x)=\left(\frac{x-5}{2x-5}\right)\\\\\bigg| \left(\frac{f}{g}\right)'=\frac{f\:'\cdot g-g'\cdot f}{g^2}\\\\f'(x)=\frac{\left(x-5\right)'\left(2x-5\right)-\left(2x-5\right)'\left(x-5\right)}{\left(2x-5\right)^2}=\\\\=\frac{1\cdot \left(2x-5\right)-2\left(x-5\right)}{\left(2x-5\right)^2}=\\\\=\frac{2x-5-2x+10}{\left(2x-5\right)^2}=\\\\=\frac{5}{\left(2x-5\right)^2}$

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$f'(x)=0\\f(x)=3x^2-9x\\\\\big |\: \left(f\pm g\right)'=f\:'\pm g'\\\\f'(x)=(3x^2-9x)'= 6x-9\\6x-9=0\\6x=9\\x=\frac{9}{6} =\frac{3}{2} =1,5$