What is (\partial)/(\partial x)(y^{9x}) ?

The answer to (\partial)/(\partial x)(y^{9x}) is 9y^{9x}ln(y)

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(\partial)/(\partial x)(y^{9x})

\frac{\partial}{\partial x} (y^{9 x})

9 y^{9 x} ln (y)

Solution steps

\frac{\partial}{\partial x} (y^{9 x})

Treat

y

as a constant

Apply exponent rule:

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

y^{9 x} = e^{9 x l n (y)}

= \frac{\partial}{\partial x} (e^{9 x l n (y)})

Apply the chain rule:

e^{9 x l n (y)} \frac{\partial}{\partial x} (9 x ln (y))

= e^{9 x l n (y)} \frac{\partial}{\partial x} (9 x ln (y))

\frac{\partial}{\partial x} (9 x ln (y)) = 9 ln (y)

= e^{9 x l n (y)} \cdot 9 ln (y)

e^{9 x l n (y)} = y^{9 x}

= 9 y^{9 x} ln (y)

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What is (\partial)/(\partial x)(y^{9x}) ?
The answer to (\partial)/(\partial x)(y^{9x}) is 9y^{9x}ln(y)