What is (\partial)/(\partial x)(ycos(pix^2y)) ?

The answer to (\partial)/(\partial x)(ycos(pix^2y)) is -2piy^2xsin(piyx^2)

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(\partial)/(\partial x)(ycos(pix^2y))

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

- 2 π y^{2} x sin (π y x^{2})

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

= y (- sin (π x^{2} y) \cdot 2 π y x)

Simplify

y (- sin (π x^{2} y) \cdot 2 π y x) : - 2 π y^{2} x sin (π y x^{2})

= - 2 π y^{2} x sin (π y x^{2})

What is (\partial)/(\partial x)(ycos(pix^2y)) ?
The answer to (\partial)/(\partial x)(ycos(pix^2y)) is -2piy^2xsin(piyx^2)