What is (\partial)/(\partial x)(cos(xy)e^{x^2}) ?

The answer to (\partial)/(\partial x)(cos(xy)e^{x^2}) is -e^{x^2}ysin(xy)+2e^{x^2}xcos(xy)

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(\partial)/(\partial x)(cos(xy)e^{x^2})

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

- e^{x^{2}} y sin (x y) + 2 e^{x^{2}} x cos (x y)

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = cos (x y), g = e^{x^{2}}

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

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

\frac{\partial}{\partial x} (e^{x^{2}}) = e^{x^{2}} \cdot 2 x

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

Simplify

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

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What is (\partial)/(\partial x)(cos(xy)e^{x^2}) ?
The answer to (\partial)/(\partial x)(cos(xy)e^{x^2}) is -e^{x^2}ysin(xy)+2e^{x^2}xcos(xy)