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Step-by-step Solution

Find the derivative using the quotient rule (d/dx)((x^2y^2)/(x^4+y^4))

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Answer

$\frac{2xy^2x^4+2xy^{6}-4y^2x^{5}}{\left(x^4+y^4\right)^2}$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{x^2 y^2}{x^4+y^4}\right)$
1

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\left(x^4+y^4\right)\frac{d}{dx}\left(x^2y^2\right)-x^2y^2\frac{d}{dx}\left(x^4+y^4\right)}{\left(x^4+y^4\right)^2}$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{y^2\left(x^4+y^4\right)\frac{d}{dx}\left(x^2\right)-x^2y^2\frac{d}{dx}\left(x^4+y^4\right)}{\left(x^4+y^4\right)^2}$

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Answer

$\frac{2xy^2x^4+2xy^{6}-4y^2x^{5}}{\left(x^4+y^4\right)^2}$
$\frac{d}{dx}\left(\frac{x^2 y^2}{x^4+y^4}\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

~ 0.97 seconds

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