What is (\partial)/(\partial y)(ycos(xln(y))) ?

The answer to (\partial)/(\partial y)(ycos(xln(y))) is cos(xln(y))-xsin(xln(y))

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(\partial)/(\partial y)(ycos(xln(y)))

\frac{\partial}{\partial y} (y cos (x ln (y)))

cos (x ln (y)) - x sin (x ln (y))

Solution steps

\frac{\partial}{\partial y} (y cos (x ln (y)))

Treat

x

as a constant

Apply the Product Rule:

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

f = y, g = cos (x ln (y))

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

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

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

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

Simplify

1 \cdot cos (x ln (y)) + (- \frac{x sin ( x ln ( y ) )}{y}) y : cos (x ln (y)) - x sin (x ln (y))

= cos (x ln (y)) - x sin (x ln (y))

\int_{0}^{2} 1 + 3 x^{2} d x

6 x^{- 3}, at x = 5

\frac{x - y}{x + y}

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

\int_{0}^{1} 2 x^{2} - 1 d x

What is (\partial)/(\partial y)(ycos(xln(y))) ?
The answer to (\partial)/(\partial y)(ycos(xln(y))) is cos(xln(y))-xsin(xln(y))