Question

Знайти похідну(производную)

Answer 1

$\displaystyle\bf\\1)\\\\y=x^{2} (x^{2} +3)\\\\y'=(x^{2} )'\cdot(x^{2} +3)+x^{2} \cdot(x^{2} +3)'=2x\cdot(x^{2} +3)+x^{2} \cdot 2x=\\\\=2x^{3} +6x+2x^{3}=4x^{3}+6x\\\\\\2)\\\\y=Sinx(x-3)\\\\y'=(Sinx)'\cdot(x-3)+Sinx\cdot(x-3)'=Cosx\cdot(x-3)+Sinx\\\\3)\\\\y=\frac{5Sinx}{x^{2} -4} \\\\\\y'=\frac{5\cdot(sinx)'\cdot(x^{2} -4)-5Sinx\cdot(x^{2} -4)'}{(x^{2} -4)^{2} } =\\\\\\=\frac{5Cosx\cdot(x^{2} -4)-10xSinx}{(x^{2} -4)^{2} }\\\\\\4)\\\\f(x)=2x^{3} -x$

$\displaystyle\bf\\f'(x)=2\cdot(x^{3})'-x'=2\cdot 3x^{2} -1=6x^{2} -1 \\\\f'(x)=0\\\\6x^{2} -1=0\\\\6x^{2} =1\\\\x^{2} =\frac{1}{6} \\\\x_{1,2} =\pm \ \sqrt{\frac{1}{6} } =\pm \ \frac{1}{\sqrt{6} } =\pm \ \frac{\sqrt{6} }{6}$