Question

derivada implicita de y^4-y=x^3

Answer 1

Derivada implícita quando derivar o y multiplique por y' ( regra da cadeia ) e no final isola y'  

$\text y^4-\text y = \text x ^3$

Derivando :

$4.\text y^3\text y'-\text y' = 3.\text x ^2$

$\text y'.(4.\text y^3-1) = 3.\text x ^2$

$\displaystyle \huge\boxed{\text y' = \frac{3\text x^2}{4\text y^3-1}}$

Answer 2

✅ Tendo finalizado os cálculos, concluímos que a derivada implícita da curva definida implicitamente em termos da incógnita "x" é:

                  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf y' = \frac{3x^{2}}{4y^{3} - 1}\:\:\:}}\end{gathered} }$

 

Seja a função definida implicitamente:

                          $\Large\displaystyle\begin{gathered} y^{4} - y = x^{3}\end{gathered}$

Calculando a derivada implícita, temos da incógnita "x":

   $\Large \text { \begin{aligned}(y^{4} - y)' & = (x^{3})'\\4\cdot y^{4 - 1}y' - 1\cdot y^{1 - 1}y' & = 3\cdot x^{3 - 1}\\4y^{3}y' - 1\cdot y^{0}y' & = 3x^{2}\\ 4y^{3}y' - 1\cdot 1\cdot y' & = 3x^{2}\\4y^{3}y' - y' & = 3x^{2}\\(4y^{3} - 1)y' & = 3x^{2}\\y' & = \frac{3x^{2}}{4y^{3} - 1}\end{aligned} }$

✅ Portanto, a derivada implícita em termos da incógnita "x" é:

                                       $\Large\displaystyle\text{ \begin{gathered} y' = \frac{3x^{2}}{4y^{3} - 1}\end{gathered} }$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$