Question

y=(x+3)³ + (x-3)² , qual a seria a derivada de y' + y'' ?

Answer 1

$y=(x+3)^{3}+(x-3)^{2}$

Derivando y (regra da cadeia):

$y'=3\cdot(x+3)^{3-1}\frac{d}{dx}(x+3)+2\cdot(x-3)^{2-1}\frac{d}{dx}(x-3)\\\\y'=3(x+3)^{2}\cdot1+2(x-3)\cdot1\\\\\boxed{\boxed{y'=3(x+3)^{2}+2x-6}}$

Derivando y' (segunda derivada, usando a regra da cadeia também):

$y''=3\cdot2(x+3)^{2-1}\frac{d}{dx}(x+3)+2\cdot1x^{1-1}-0\\\\y''=6(x+3)^{1}\cdot1+2\\\\y''=6x+18+2\\\\\boxed{\boxed{y''=6x+20}}$

Então:

$y'+y''=3(x+3)^{2}+2x-6+6x+20\\\\\boxed{\boxed{y'+y''=3(x+3)^{2}+8x+14}}$