Pregunta
Siven the equation \( \sqrt{x^{2}+4 x y+y^{6}}=8 \), evaluate \( \frac{d y}{d x} \). Assume that the equation implicitly defines \( y \) as a differentiable \( F(x, y)=\sqrt{x^{2}+4 x y+y^{6}}-8=0 \), then \( F_{x}=\frac{x+2 y}{\sqrt{x^{2}+4 x y+y^{6}}} \) \( F(x, y)=\sqrt{x^{2}+4 x y+y^{6}}-8=0 \), then \( F_{y}=\square \)
Ask by French Chen. in the United States
Mar 17,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( F_y = \frac{2x + 3y^{5}}{\sqrt{x^{2} + 4xy + y^{6}}} \)
Solución
We start with the function
\[
F(x, y)=\sqrt{x^{2}+4xy+y^{6}}-8=0.
\]
The partial derivative with respect to \(y\) is computed by applying the chain rule. Let
\[
u=x^{2}+4xy+y^{6}.
\]
Then
\[
F(x, y)=\sqrt{u}-8.
\]
Taking the derivative with respect to \(y\):
\[
F_y=\frac{d}{dy}\left(\sqrt{u}\right)
=\frac{1}{2\sqrt{u}}\,\frac{du}{dy}.
\]
Now, we find \(\frac{du}{dy}\). Differentiating \(u\) with respect to \(y\):
\[
\frac{du}{dy}=0+4x+6y^{5}=4x+6y^{5}.
\]
Therefore, we have
\[
F_y=\frac{4x+6y^{5}}{2\sqrt{x^{2}+4xy+y^{6}}}
=\frac{2x+3y^{5}}{\sqrt{x^{2}+4xy+y^{6}}}.
\]
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Simplifique esta solución Beyond the Answer
To find \( F_y \), we need to differentiate \( F(x, y) \) with respect to \( y \). The function \( F(x, y) \) is defined as: \[ F(x, y) = \sqrt{x^2 + 4xy + y^6} - 8 \] Now, taking the derivative of \( F \) with respect to \( y \): \[ F_y = \frac{\partial}{\partial y} \left( \sqrt{x^2 + 4xy + y^6} \right) \] Using the chain rule: \[ F_y = \frac{1}{2\sqrt{x^2 + 4xy + y^6}} \cdot (4x + 6y^5) \] Hence, we can summarize this as: \[ F_y = \frac{4x + 6y^5}{2\sqrt{x^2 + 4xy + y^6}} = \frac{2x + 3y^5}{\sqrt{x^2 + 4xy + y^6}} \] So, \( F_y \) evaluates to \( \frac{2x + 3y^5}{\sqrt{x^2 + 4xy + y^6}} \).
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