What is (\partial)/(\partial x)(xcos(yln(x))) ?

The answer to (\partial)/(\partial x)(xcos(yln(x))) is cos(yln(x))-ysin(yln(x))

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(\partial)/(\partial x)(xcos(yln(x)))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

x \to 0.1 lim (\frac{1}{x})

\int \frac{1}{( 81 - x ^{2} ) ^{\frac{3}{2}}} d x

\int cos (\frac{x}{2}) \cdot cos (n x) d x

\int t^{2} e^{t} d t

n = 1 \sum \infty \frac{n - 1}{n}

What is (\partial)/(\partial x)(xcos(yln(x))) ?
The answer to (\partial)/(\partial x)(xcos(yln(x))) is cos(yln(x))-ysin(yln(x))