What is (\partial)/(\partial x)(y^{x^7}) ?

The answer to (\partial)/(\partial x)(y^{x^7}) is 7x^6y^{x^7}ln(y)

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(\partial)/(\partial x)(y^{x^7})

\frac{\partial}{\partial x} (y^{x^{7}})

7 x^{6} y^{x^{7}} ln (y)

Solution steps

\frac{\partial}{\partial x} (y^{x^{7}})

Treat

y

as a constant

Apply exponent rule:

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

y^{x^{7}} = e^{x^{7} l n (y)}

= \frac{\partial}{\partial x} (e^{x^{7} l n (y)})

Apply the chain rule:

e^{x^{7} l n (y)} \frac{\partial}{\partial x} (x^{7} ln (y))

= e^{x^{7} l n (y)} \frac{\partial}{\partial x} (x^{7} ln (y))

\frac{\partial}{\partial x} (x^{7} ln (y)) = 7 x^{6} ln (y)

= e^{x^{7} l n (y)} \cdot 7 x^{6} ln (y)

e^{x^{7} l n (y)} = y^{x^{7}}

= 7 x^{6} y^{x^{7}} ln (y)

What is (\partial)/(\partial x)(y^{x^7}) ?
The answer to (\partial)/(\partial x)(y^{x^7}) is 7x^6y^{x^7}ln(y)