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

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

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

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

8 y^{8 x} ln (y)

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

y^{8 x} = e^{8 x l n (y)}

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (8 x ln (y)) = 8 ln (y)

= e^{8 x l n (y)} \cdot 8 ln (y)

e^{8 x l n (y)} = y^{8 x}

= 8 y^{8 x} ln (y)

x \to \frac{1}{2} lim (\frac{x ^{2}}{x - 1})

y = tan^{2} (9 x)

x \frac{d y}{d x} + y = x

\int (x e^{3 x}) d x

\int \frac{1}{x ln ( 4 x )} d x

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