What is (\partial)/(\partial x)((x^2+y^2)^{4/5}) ?

The answer to (\partial)/(\partial x)((x^2+y^2)^{4/5}) is (8x)/(5(x^2+y^2)^{1/5)}

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(\partial)/(\partial x)((x^2+y^2)^{4/5})

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{\frac{4}{5}})

\frac{8 x}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}}

Solution steps

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{\frac{4}{5}})

Treat

y

as a constant

Apply the chain rule:

\frac{4}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}} \frac{\partial}{\partial x} (x^{2} + y^{2})

= \frac{4}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}} \frac{\partial}{\partial x} (x^{2} + y^{2})

\frac{\partial}{\partial x} (x^{2} + y^{2}) = 2 x

= \frac{4}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}} \cdot 2 x

Simplify

\frac{4}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}} \cdot 2 x : \frac{8 x}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}}

= \frac{8 x}{5 ( x ^{2} + y ^{2} ) ^{\frac{1}{5}}}

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What is (\partial)/(\partial x)((x^2+y^2)^{4/5}) ?
The answer to (\partial)/(\partial x)((x^2+y^2)^{4/5}) is (8x)/(5(x^2+y^2)^{1/5)}