What is (\partial)/(\partial x)(-ysin(xln(y)ln(y))) ?

The answer to (\partial)/(\partial x)(-ysin(xln(y)ln(y))) is -ycos(xln^2(y))ln^2(y)

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(\partial)/(\partial x)(-ysin(xln(y)ln(y)))

\frac{\partial}{\partial x} (- y sin (x ln (y) ln (y)))

- y cos (x ln^{2} (y)) ln^{2} (y)

Solution steps

\frac{\partial}{\partial x} (- y sin (x ln (y) ln (y)))

Simplify

- y sin (x ln (y) ln (y)) : - y sin (x ln^{2} (y))

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

= - y cos (x ln^{2} (y)) ln^{2} (y)

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What is (\partial)/(\partial x)(-ysin(xln(y)ln(y))) ?
The answer to (\partial)/(\partial x)(-ysin(xln(y)ln(y))) is -ycos(xln^2(y))ln^2(y)