What is (\partial)/(\partial x)(sin(xy)ln(x+y)) ?

The answer to (\partial)/(\partial x)(sin(xy)ln(x+y)) is ycos(xy)ln(x+y)+(sin(xy))/(x+y)

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(\partial)/(\partial x)(sin(xy)ln(x+y))

\frac{\partial}{\partial x} (sin (x y) ln (x + y))

y cos (x y) ln (x + y) + \frac{sin ( x y )}{x + y}

Solution steps

\frac{\partial}{\partial x} (sin (x y) ln (x + y))

Treat

y

as a constant

Apply the Product Rule:

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

f = sin (x y), g = ln (x + y)

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

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

\frac{\partial}{\partial x} (ln (x + y)) = \frac{1}{x + y}

= cos (x y) y ln (x + y) + \frac{1}{x + y} sin (x y)

\frac{1}{x + y} sin (x y) = \frac{sin ( x y )}{x + y}

= y cos (x y) ln (x + y) + \frac{sin ( x y )}{x + y}

\int (1 - y^{2})^{2} d y

y = \frac{1}{( 7 x + 12 ) ^{3}}

\frac{d}{d x} (\frac{1}{3} x^{3} + x^{2} - x + 5)

n \to \infty lim (sin (\frac{nπ}{2}))

What is (\partial)/(\partial x)(sin(xy)ln(x+y)) ?
The answer to (\partial)/(\partial x)(sin(xy)ln(x+y)) is ycos(xy)ln(x+y)+(sin(xy))/(x+y)