What is (\partial)/(\partial x)(x*cos(pi*x*y^2)) ?

The answer to (\partial)/(\partial x)(x*cos(pi*x*y^2)) is cos(pixy^2)-pixy^2sin(pixy^2)

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(\partial)/(\partial x)(xcos(pix*y^2))

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

cos (π x y^{2}) - π x y^{2} sin (π x y^{2})

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = cos (π x y^{2})

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

\frac{\partial}{\partial x} (x) = 1

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

= 1 \cdot cos (π x y^{2}) + (- sin (π x y^{2}) π y^{2}) x

Simplify

= cos (π x y^{2}) - π x y^{2} sin (π x y^{2})

\frac{\partial}{\partial t} (3 sin (t))

y^{^{'}} + \frac{2}{t} y = 4 t

\int \frac{1}{x} 1 + \frac{1}{x ^{4}} d x

y^{^{''}} + 4 y^{^{'}} + 3 y = 32 x e^{x}

\int x^{5} cos (x) d x

What is (\partial)/(\partial x)(x*cos(pi*x*y^2)) ?
The answer to (\partial)/(\partial x)(x*cos(pi*x*y^2)) is cos(pixy^2)-pixy^2sin(pixy^2)