What is (\partial)/(\partial y)(y^x-ln(x^4y+7x)) ?

The answer to (\partial)/(\partial y)(y^x-ln(x^4y+7x)) is xy^{x-1}-(x^3)/(x^3y+7)

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(\partial)/(\partial y)(y^x-ln(x^4y+7x))

\frac{\partial}{\partial y} (y^{x} - ln (x^{4} y + 7 x))

x y^{x - 1} - \frac{x ^{3}}{x ^{3} y + 7}

Solution steps

\frac{\partial}{\partial y} (y^{x} - ln (x^{4} y + 7 x))

Treat

x

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial y} (y^{x}) - \frac{\partial}{\partial y} (ln (x^{4} y + 7 x))

\frac{\partial}{\partial y} (y^{x}) = x y^{x - 1}

\frac{\partial}{\partial y} (ln (x^{4} y + 7 x)) = \frac{x ^{3}}{x ^{3} y + 7}

= x y^{x - 1} - \frac{x ^{3}}{x ^{3} y + 7}

What is (\partial)/(\partial y)(y^x-ln(x^4y+7x)) ?
The answer to (\partial)/(\partial y)(y^x-ln(x^4y+7x)) is xy^{x-1}-(x^3)/(x^3y+7)