What is (\partial)/(\partial x)(z^{x^y}) ?

The answer to (\partial)/(\partial x)(z^{x^y}) is yz^{x^y}x^{y-1}ln(z)

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(\partial)/(\partial x)(z^{x^y})

\frac{\partial}{\partial x} (z^{x^{y}})

y z^{x^{y}} x^{y - 1} ln (z)

Solution steps

\frac{\partial}{\partial x} (z^{x^{y}})

Treat

z, y

as constants

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (x^{y} ln (z)) = ln (z) y x^{y - 1}

= e^{x^{y} l n (z)} ln (z) y x^{y - 1}

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

= y z^{x^{y}} x^{y - 1} ln (z)

f (x) = \frac{x}{x ^{2} + 81}

x = 2 y^{2} - 5, x = y^{2} + 4

y = 2 x - 5

\int x^{- 12} d x

f (x) = 6 x^{3} - 9 x + 4

What is (\partial)/(\partial x)(z^{x^y}) ?
The answer to (\partial)/(\partial x)(z^{x^y}) is yz^{x^y}x^{y-1}ln(z)