What is (\partial)/(\partial x)(5xsin(xy)) ?

The answer to (\partial)/(\partial x)(5xsin(xy)) is 5(sin(xy)+xycos(xy))

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(\partial)/(\partial x)(5xsin(xy))

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

5 (sin (x y) + x y cos (x y))

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x, g = sin (x y)

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

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

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

= 5 (1 \cdot sin (x y) + cos (x y) y x)

Simplify

= 5 (sin (x y) + x y cos (x y))

\frac{\partial}{\partial x} (ln (8 y e^{x y}))

\int \frac{1}{( 25 - x ^{2} ) ^{2}} d x

3 x^{2} - x^{3}

\frac{\partial}{\partial x} (3 x^{2} + y^{2})

\int_{x}^{3} sin (t^{2}) d t

What is (\partial)/(\partial x)(5xsin(xy)) ?
The answer to (\partial)/(\partial x)(5xsin(xy)) is 5(sin(xy)+xycos(xy))