What is the derivative of x^y ?

The derivative of x^y is x^y(ln(x)(dy)/(dx)+y/x)

What is the first derivative of x^y ?

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

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

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

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{y} = e^{y l n (x)}

= \frac{d}{d x} (e^{y l n (x)})

Apply the chain rule:

e^{y l n (x)} \frac{d}{d x} (y ln (x))

= e^{y l n (x)} \frac{d}{d x} (y ln (x))

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

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

e^{y l n (x)} = x^{y}

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

(x^{2} + 1) (x^{2} - 1) (x^{2} + 1)

x^{2} \cdot e^{- x}

f (x) = \frac{e ^{x}}{e ^{3 x}}

(6 x^{2}) (4 x^{2})

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