Question

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Answer 1

1) y = - 15x² + 3x
    y ' = - 30x
    y'(5) = - 30 * 5 = - 150
2) y = $\frac{4 \sqrt{x} }{ e^{x} }$
    y' = $\frac{(4 \sqrt{x})' * e^{x} - (4\sqrt{x}) * ( e^{x})' }{( e^{x}) {2} } = \frac{ \frac{2}{ \sqrt{x} }* e^{x} - 4\sqrt{x} * e^{x} }{( e^{x}) ^{2} } = \frac{( \frac{2}{ \sqrt{x} }- 4\sqrt{x} ) }{ e^{x} }$
y'(1) = $\frac{ \frac{2}{ \sqrt{1} }- 4\sqrt{1} }{ e^{1} } = - \frac{2}{e}$
3) y = (x + 1)* tgx
    y' = (x+1)' * tgx + (x+1) * (tgx)' = tgx + (x+1) * $\frac{1}{ Cos^{2}x }$
    y'(π/4) = tg π/4 + (π/4 + 1) * $\frac{1}{ Cos^{2} \pi /4 }$ = 1 +
+ 2(π/4 + 1) = 1 + π/2 + 2 = 3 + π/2
4) y = (Sinx+3)(Cosx-1)
    y' = (Sinx+3)'(Cosx-1)+(Sinx+3)(Cosx-1)' = Cosx(Cosx-1) - (Sinx+3)Sinx
    y'(π/6) = Cos π/6(Cos π/6 - 1) - (Sin π/6 + 3)Sin π/6 = $\frac{ \sqrt{3} }{2}$($\frac{ \sqrt{3} }{2}$- 1) - ($\frac{1}{2}$+ 3)*$\frac{1}{2}$ = $\frac{3}{4 } - \frac{ \sqrt{3} }{2} - \frac{7}{4} = - (1 + \frac{ \sqrt{3} }{2} )$