What is the derivative of x^{y^x} ?

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

What is the first derivative of x^{y^x} ?

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\int \frac{3}{x ^{2} + 4} d x

\int x^{4} x^{5} + 3 d x

\frac{\partial}{\partial y} (2 e^{\frac{x}{2} + y} - y)

\frac{490 t}{( t ^{2} + 7 ) ^{3}}

\frac{d}{d x} (\frac{x ^{2} ( x + 2 ) ^{4}}{( 2 x ^{2} - 1 ) ^{3}})

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