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

The answer to (\partial)/(\partial x)(e^{2x}cos(x^2y)) is 2e^{2x}cos(x^2y)-2e^{2x}xysin(x^2y)

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

x \to - 5 + lim (\frac{2}{x ^{2} - 25})

\frac{d}{d x} (x^{3} (x - 4)^{2})

f (x) = \frac{2 x}{( x ^{2} - x )}

\frac{d}{d x} (\frac{1}{3 x})

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What is (\partial)/(\partial x)(e^{2x}cos(x^2y)) ?
The answer to (\partial)/(\partial x)(e^{2x}cos(x^2y)) is 2e^{2x}cos(x^2y)-2e^{2x}xysin(x^2y)