Question

e="F(x)=(6 \sqrt{x} -2)(5 \sqrt{x} +7)" alt="F(x)=(6 \sqrt{x} -2)(5 \sqrt{x} +7)" align="absmiddle" class="latex-formula">

Answer 1

$\bf f(x)=(6\sqrt{x}-2)(5\sqrt{x}+7) \\\\\\ \cfrac{dy}{dx}=\stackrel{product~rule}{\left( 6\cdot \frac{1}{2}x^{-\frac{1}{2}} \right)(5x^{\frac{1}{2}}+7)~~+~~(6x^{\frac{1}{2}}-2)\left(5\cdot \frac{1}{2}x^{-\frac{1}{2}} \right)} \\\\\\ \cfrac{dy}{dx}=\left(\cfrac{6}{2}\cdot \cfrac{1}{\sqrt{x}} \right)(5x^{\frac{1}{2}}+7)~~+~~2(3x^{\frac{1}{2}}-1)\left(\cfrac{5}{2}x^{-\frac{1}{2}} \right)$

$\bf \cfrac{dy}{dx}=\left(3\cdot \cfrac{1}{\sqrt{x}} \right)(5x^{\frac{1}{2}}+7)~~+~~2\cdot \cfrac{5}{2}(3x^{\frac{1}{2}}-1)\left(\cfrac{1}{\sqrt{x}} \right) \\\\\\ \cfrac{dy}{dx}=\cfrac{3(5\sqrt{x}+7)}{\sqrt{x}}~~+~~\cfrac{5(3\sqrt{x}-1)}{\sqrt{x}}\\\\\\ \cfrac{dy}{dx}=\cfrac{15\sqrt{x}+21~~+~~15\sqrt{x}-5}{\sqrt{x}} \\\\\\ \cfrac{dy}{dx}=\cfrac{30\sqrt{x}+16}{\sqrt{x}}$