Question

derivada b) y = ( 1 + 3√x)^3
c) y = 2x^2 - 1/ x √1+x^2

Answer 1

b) $\frac{d}{dx} (1+3 \sqrt{x} ) ^{3}$

1º Passo: derivar o parenteses
$<strong>3(1+3 \sqrt{x} ) ^{2}</strong> \frac{d}{dx} (1+3 \sqrt{x} )$

2º Passo: derivar o que estava dentro do parenteses
$3(1+3 \sqrt{x} )^{2} <strong>( \frac{d}{dx}1 + 3\frac{d}{dx} \sqrt{x} )</strong>$

$3(1+3 \sqrt{x} )^{2}(0 + 3 \frac{d}{dx} x^{ \frac{1}{2} } )= 3(1+3 \sqrt{x} )^{2}3( \frac{1}{2 \sqrt{x} })$


$\frac{9(1+3 \sqrt{x} )^{2} }{2 \sqrt{x} } = \frac{9(1 + 6 \sqrt{x} + 9x) \sqrt{x} }{2x} = \frac{9 \sqrt{x} + 54x + 81x \sqrt{x} }{2x}$


c) $\frac{d}{dx}(2 x^{2} - \frac{1}{x \sqrt{1+ x^{2} } } )$

$2 \frac{d}{dx}(x^{2} )- \frac{d}{dx}( \frac{1}{x \sqrt{1+ x^{2} } } )= 4x - (- \frac{ \frac{d}{dx}(x \sqrt{1+ x^{2} }) }{(x \sqrt{1+ x^{2} })² } )$

$4x-(- \frac{ (\frac{d}{dx}x) \sqrt{1+ x^{2} }+x \frac{d}{dx} \sqrt{1 + x^{2} } }{(x \sqrt{1+ x^{2} } )^{2} }) = 4x-( \frac{ \sqrt{1+ x^{2} }+x \frac{1}{2 \sqrt{1+ x^{2} } }2x }{(x \sqrt{1+ x^{2} } )^{2} } )$

$\frac{4x \sqrt{1+ x^{2} }( x^{2} + x^{4}) +1+2 x^{2} }{ \sqrt{1+ x^{2} }( x^{2} + x^{4}) }$