Question

Calcule a derivada de cada função:

a)f(x)=2/3 x³
b)f(x)=3 raiz cúbica x² + 5
c)f(x)=x³ + raiz quinta x³
d)f(x)=1 sobre 4 x² + 1 sobre 9 x³

Answer 1

$f(x) = \frac{2}{3*x^3} = \frac{2}{3}* x^{-3}$

mantem a constante 2/3 e deriva x^(-3)

$f'(x) = \frac{2}{3}* -3*(x^{-3-1} )\\\\f'(x) = \frac{2*-3}{3}*(x^{-4}) \\\\f'(x) = 2* (-1) * x^{-4}\\\\\boxed{f'(x) = \frac{-2}{x^4} }$
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$f(x) = 3 \sqrt[3]{x^2+5} \\\\f(x) = 3* (x^2+5)^ \frac{1}{3}$

mantem a constante e usa a regra da cadeia para derivar 
$\boxed{u^n = n* u^{n-1} * u'}$

 $u=x^2+5\\\ u'= 2x$

substituindo na formula
$\not 3* \frac{1}{\not3}(x^2+5) ^{( \frac{1}{3} -1) } * 2x\\\\(x^2+5)^{ \frac{-2}{3} } * 2x\\\\ \frac{1}{(x^2+5y)^ \frac{2}{3} } *2x\\\\ \boxed{\boxed{f'(x) = \frac{2x}{ \sqrt[3]{(x^2+5)^2} } }}$
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$f(x) = x^3 + \sqrt[5]{x^3} = x^3 + x^ \frac{3}{5}$

derivando x³ temos 3x² 

derivando
 $x^ \frac{1}{5} = \frac{1}{5}* x^{ \frac{1}{5}-1 } \\\\= \frac{1}{5}* x^{ \frac{-4}{5}}\\\\ = \frac{1}{5}* \frac{1}{x^{ \frac{4}{5} }} }\\\\ = \frac{1}{5}* \frac{1}{ \sqrt[5]{x^4} } =\boxed{ \frac{1}{5* \sqrt[5] {x^4} } }$

derivada de uma soma e a soma das derivadas
$\boxed{f'(x) = 3x^2 + \frac{1}{5 \sqrt[5] {x^4} }}$

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$f(x) = \frac{1}{4x^2 } + \frac{1}{9x^3 } \\\\f(x) = (4x^2)^{-1} + (9x^3)^{-1}$

derivando
(4x²)^(-1) 
utilizando a regra da cadeia

u = 4x²
u' = 8x

$-1*(4x^2)^{-1-1} * 8x\\\\ -1* \frac{1}{(4x^2)^2} * 8x\\\\ \frac{-8x}{4^2 *(x^2)^2}= \frac{-8x}{16x^4}= \frac{-1}{2x^3}$

aplicando a mesma coisa para (9x³)^(-1)

u = 9x³
u' = 27x²
$-1*(9x^3)^{-2} * 27x^2\\\\= \frac{-27x^2}{(9x^3)^2}= \frac{-27x^2}{81x^6} = \frac{-1}{3x^4}$

somando as duas derivadas
$f'(x) = \frac{-1}{2x^3}- \frac{1}{3x^4}$