What is (\partial)/(\partial x)([Ax^2+By^3]^{1/4}) ?

The answer to (\partial)/(\partial x)([Ax^2+By^3]^{1/4}) is (Ax)/(2(Ax^2+By^3)^{3/4)}

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(\partial)/(\partial x)([Ax^2+By^3]^{1/4})

\frac{\partial}{\partial x} ([A x^{2} + B y^{3}]^{\frac{1}{4}})

\frac{A x}{2 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}}

Solution steps

\frac{\partial}{\partial x} ((A x^{2} + B y^{3})^{\frac{1}{4}})

Treat

A, B, y

as a constant

Apply the chain rule:

\frac{1}{4 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}} \frac{\partial}{\partial x} (A x^{2} + B y^{3})

= \frac{1}{4 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}} \frac{\partial}{\partial x} (A x^{2} + B y^{3})

\frac{\partial}{\partial x} (A x^{2} + B y^{3}) = 2 A x

= \frac{1}{4 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}} \cdot 2 A x

Simplify

\frac{1}{4 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}} \cdot 2 A x : \frac{A x}{2 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}}

= \frac{A x}{2 ( A x ^{2} + B y ^{3} ) ^{\frac{3}{4}}}

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\int \frac{1}{4 x y ^{2}} d x

What is (\partial)/(\partial x)([Ax^2+By^3]^{1/4}) ?
The answer to (\partial)/(\partial x)([Ax^2+By^3]^{1/4}) is (Ax)/(2(Ax^2+By^3)^{3/4)}