What is (\partial)/(\partial x)(yln(x^4+y^3+1)) ?

The answer to (\partial)/(\partial x)(yln(x^4+y^3+1)) is (4yx^3)/(x^4+y^3+1)

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(\partial)/(\partial x)(yln(x^4+y^3+1))

\frac{\partial}{\partial x} (y ln (x^{4} + y^{3} + 1))

\frac{4 y x ^{3}}{x ^{4} + y ^{3} + 1}

Solution steps

\frac{\partial}{\partial x} (y ln (x^{4} + y^{3} + 1))

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= y \frac{\partial}{\partial x} (ln (x^{4} + y^{3} + 1))

Apply the chain rule:

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

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

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

= y \frac{1}{x ^{4} + y ^{3} + 1} \cdot 4 x^{3}

Simplify

y \frac{1}{x ^{4} + y ^{3} + 1} \cdot 4 x^{3} : \frac{4 y x ^{3}}{x ^{4} + y ^{3} + 1}

= \frac{4 y x ^{3}}{x ^{4} + y ^{3} + 1}

What is (\partial)/(\partial x)(yln(x^4+y^3+1)) ?
The answer to (\partial)/(\partial x)(yln(x^4+y^3+1)) is (4yx^3)/(x^4+y^3+1)