What is (\partial)/(\partial x)(xsin(5x^2y)) ?

The answer to (\partial)/(\partial x)(xsin(5x^2y)) is sin(5x^2y)+10x^2ycos(5x^2y)

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(\partial)/(\partial x)(xsin(5x^2y))

\frac{\partial}{\partial x} (x sin (5 x^{2} y))

sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y)

Solution steps

\frac{\partial}{\partial x} (x sin (5 x^{2} y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = sin (5 x^{2} y)

= \frac{\partial}{\partial x} (x) sin (5 x^{2} y) + \frac{\partial}{\partial x} (sin (5 x^{2} y)) x

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

\frac{\partial}{\partial x} (sin (5 x^{2} y)) = cos (5 x^{2} y) \cdot 10 y x

= 1 \cdot sin (5 x^{2} y) + cos (5 x^{2} y) \cdot 10 y xx

Simplify

1 \cdot sin (5 x^{2} y) + cos (5 x^{2} y) \cdot 10 y xx : sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y)

= sin (5 x^{2} y) + 10 x^{2} y cos (5 x^{2} y)

\int - 2 sin^{5} (x) cos^{2} (x) d x

\frac{d}{d x} (1.795 x^{2})

t^{3} \frac{d x}{d t} + 3 t^{2} x = t

y = x^{2} e^{- x}, (1, \frac{1}{e})

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

What is (\partial)/(\partial x)(xsin(5x^2y)) ?
The answer to (\partial)/(\partial x)(xsin(5x^2y)) is sin(5x^2y)+10x^2ycos(5x^2y)