What is (\partial)/(\partial x)(ln(x^8y)) ?

The answer to (\partial)/(\partial x)(ln(x^8y)) is 8/x

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(\partial)/(\partial x)(ln(x^8y))

\frac{\partial}{\partial x} (ln (x^{8} y))

\frac{8}{x}

Solution steps

\frac{\partial}{\partial x} (ln (x^{8} y))

Treat

y

as a constant

Apply the chain rule:

\frac{1}{x ^{8} y} \frac{\partial}{\partial x} (x^{8} y)

= \frac{1}{x ^{8} y} \frac{\partial}{\partial x} (x^{8} y)

\frac{\partial}{\partial x} (x^{8} y) = 8 y x^{7}

= \frac{1}{x ^{8} y} \cdot 8 y x^{7}

Simplify

\frac{1}{x ^{8} y} \cdot 8 y x^{7} : \frac{8}{x}

= \frac{8}{x}

x \to 0 lim (- \frac{1}{x ^{2}})

\frac{d}{d x} (3 sin^{2} (2 x))

\int \frac{sin ^{2} ( x )}{1 + cos ^{2} ( x )} d x

\int 15 x^{3} e^{x} d x

x \to \infty lim (- x)

What is (\partial)/(\partial x)(ln(x^8y)) ?
The answer to (\partial)/(\partial x)(ln(x^8y)) is 8/x