What is (\partial)/(\partial x)(|xy|^n) ?

The answer to (\partial)/(\partial x)(|xy|^n) is nx|y|^n|x|^{n-2}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(\partial)/(\partial x)(|xy|^n)

\frac{\partial}{\partial x} (∣ x y ∣^{n})

n x ∣ y ∣^{n} ∣ x ∣^{n - 2}

Solution steps

\frac{\partial}{\partial x} (∣ x y ∣^{n})

Simplify

∣ x y ∣^{n} : (∣ x ∣ ∣ y ∣)^{n}

= \frac{\partial}{\partial x} ((∣ x ∣ ∣ y ∣)^{n})

Treat

n, y

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (∣ x ∣ ∣ y ∣) = \frac{x ∣ y ∣}{∣ x ∣}

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

Simplify

n (∣ x ∣ ∣ y ∣)^{n - 1} \frac{x ∣ y ∣}{∣ x ∣} : n x ∣ y ∣^{n} ∣ x ∣^{n - 2}

= n x ∣ y ∣^{n} ∣ x ∣^{n - 2}

\int 8 sin^{5} (x) d x

\int_{0}^{r} r^{2} - y^{2} d y

\int ln (5^{u}) d u

\frac{d}{d x} (cos (n x))

\frac{d y}{d x} = x^{3} y^{3}

What is (\partial)/(\partial x)(|xy|^n) ?
The answer to (\partial)/(\partial x)(|xy|^n) is nx|y|^n|x|^{n-2}