What is (\partial)/(\partial x)(ln(y^{x^2})) ?

The answer to (\partial)/(\partial x)(ln(y^{x^2})) is 2xln(y)

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(\partial)/(\partial x)(ln(y^{x^2}))

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

2 x ln (y)

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

= \frac{1}{y ^{x^{2}}} \cdot 2 y^{x^{2}} x ln (y)

Simplify

\frac{1}{y ^{x^{2}}} \cdot 2 y^{x^{2}} x ln (y) : 2 x ln (y)

= 2 x ln (y)

What is (\partial)/(\partial x)(ln(y^{x^2})) ?
The answer to (\partial)/(\partial x)(ln(y^{x^2})) is 2xln(y)