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Popular >

derivative of ln(x^y)

Solution

\frac{d}{d x} (ln (x^{y}))

Solution

ln (x) \frac{d y}{d x} + \frac{y}{x}

Solution steps

\frac{d}{d x} (ln (x^{y}))

Apply the chain rule:

\frac{1}{x ^{y}} \frac{d}{d x} (x^{y})

= \frac{1}{x ^{y}} \frac{d}{d x} (x^{y})

\frac{d}{d x} (x^{y}) = x^{y} (ln (x) \frac{d y}{d x} + \frac{y}{x})

= \frac{1}{x ^{y}} x^{y} (ln (x) \frac{d y}{d x} + \frac{y}{x})

Simplify

\frac{1}{x ^{y}} x^{y} (ln (x) \frac{d y}{d x} + \frac{y}{x}) : ln (x) \frac{d y}{d x} + \frac{y}{x}

= ln (x) \frac{d y}{d x} + \frac{y}{x}

Popular Examples

integral of u^{-2}(2u^4+3u^2-2u^{-3})

\int u^{- 2} (2 u^{4} + 3 u^{2} - 2 u^{- 3}) d u

derivative of (3x-4/(2x^2+5))

\frac{d}{d x} (\frac{3 x - 4}{2 x ^{2} + 5})

derivative of x^2e^{-2x}

\frac{d}{d x} (x^{2} e^{- 2 x})

limit as x approaches 1/2 of |2x+3|

x \to \frac{1}{2} lim (∣ 2 x + 3 ∣)

(\partial)/(\partial x)(x^6y^8+7x^8y)

\frac{\partial}{\partial x} (x^{6} y^{8} + 7 x^{8} y)

Frequently Asked Questions (FAQ)

What is the derivative of ln(x^y) ?
The derivative of ln(x^y) is ln(x)(dy)/(dx)+y/x
What is the first derivative of ln(x^y) ?
The first derivative of ln(x^y) is ln(x)(dy)/(dx)+y/x