What is (\partial)/(\partial x)(z^{xy^2}) ?

The answer to (\partial)/(\partial x)(z^{xy^2}) is y^2z^{xy^2}ln(z)

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(\partial)/(\partial x)(z^{xy^2})

\frac{\partial}{\partial x} (z^{x y^{2}})

y^{2} z^{x y^{2}} ln (z)

Solution steps

\frac{\partial}{\partial x} (z^{x y^{2}})

Treat

z, y

as constants

Apply exponent rule:

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

z^{x y^{2}} = e^{x y^{2} l n (z)}

= \frac{\partial}{\partial x} (e^{x y^{2} l n (z)})

Apply the chain rule:

e^{x y^{2} l n (z)} \frac{\partial}{\partial x} (x y^{2} ln (z))

= e^{x y^{2} l n (z)} \frac{\partial}{\partial x} (x y^{2} ln (z))

\frac{\partial}{\partial x} (x y^{2} ln (z)) = y^{2} ln (z)

= e^{x y^{2} l n (z)} y^{2} ln (z)

e^{x y^{2} l n (z)} = z^{x y^{2}}

= y^{2} z^{x y^{2}} ln (z)

x \to 0 + lim (\frac{x - ∣ x ∣}{∣ x ∣})

\frac{d y}{d t} = 2 t e^{- y}, y (0) = - 2

y^{^{''}} = 4 y, y (0) = 0, y^{^{'}} (0) = 1

y^{^{'}} = x (x - 1)

(y + x^{3} + x y^{2}) d x - x d y = 0

What is (\partial)/(\partial x)(z^{xy^2}) ?
The answer to (\partial)/(\partial x)(z^{xy^2}) is y^2z^{xy^2}ln(z)