What is (\partial)/(\partial x)(ln(x+y^3+xy^2)) ?

The answer to (\partial)/(\partial x)(ln(x+y^3+xy^2)) is (y^2+1)/(x+y^3+xy^2)

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(\partial)/(\partial x)(ln(x+y^3+xy^2))

\frac{\partial}{\partial x} (ln (x + y^{3} + x y^{2}))

\frac{y ^{2} + 1}{x + y ^{3} + x y ^{2}}

Solution steps

\frac{\partial}{\partial x} (ln (x + y^{3} + x y^{2}))

Treat

y

as a constant

Apply the chain rule:

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

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

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

= \frac{1}{x + y ^{3} + x y ^{2}} (y^{2} + 1)

Simplify

\frac{1}{x + y ^{3} + x y ^{2}} (y^{2} + 1) : \frac{y ^{2} + 1}{x + y ^{3} + x y ^{2}}

= \frac{y ^{2} + 1}{x + y ^{3} + x y ^{2}}

\frac{\partial}{\partial z} (x^{2} y z^{2})

\frac{d}{d x} (1 - \frac{3}{x + 7})

\frac{d}{d x} (cos (e^{3 x}))

\int_{1}^{5} 2 x + 1 d x

\frac{4 x ^{2} + 4 x + 2}{x}

What is (\partial)/(\partial x)(ln(x+y^3+xy^2)) ?
The answer to (\partial)/(\partial x)(ln(x+y^3+xy^2)) is (y^2+1)/(x+y^3+xy^2)