What is (\partial)/(\partial x)(8z^{y/x}) ?

The answer to (\partial)/(\partial x)(8z^{y/x}) is -(8yz^{y/x}ln(z))/(x^2)

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(\partial)/(\partial x)(8z^{y/x})

\frac{\partial}{\partial x} (8 z^{\frac{y}{x}})

- \frac{8 y z ^{\frac{y}{x}} ln ( z )}{x ^{2}}

Solution steps

\frac{\partial}{\partial x} (8 z^{\frac{y}{x}})

Treat

z, y

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{\partial}{\partial x} (z^{\frac{y}{x}})

Apply exponent rule:

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

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

= 8 \frac{\partial}{\partial x} (e^{\frac{y}{x} l n (z)})

Apply the chain rule:

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

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

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

= 8 e^{\frac{y}{x} l n (z)} (- \frac{y ln ( z )}{x ^{2}})

Simplify

8 e^{\frac{y}{x} l n (z)} (- \frac{y ln ( z )}{x ^{2}}) : - \frac{8 y z ^{\frac{y}{x}} ln ( z )}{x ^{2}}

= - \frac{8 y z ^{\frac{y}{x}} ln ( z )}{x ^{2}}

\frac{d y}{d x} = x + 6 y, y (0) = 7

5 x y \frac{d y}{d x} = 2

\int \frac{1}{( x + 4 ) ^{5}} d x

\int x x + 7 d x

x \to - 2 + lim (x)

What is (\partial)/(\partial x)(8z^{y/x}) ?
The answer to (\partial)/(\partial x)(8z^{y/x}) is -(8yz^{y/x}ln(z))/(x^2)