What is (\partial)/(\partial x)(xsin(8/x)) ?

The answer to (\partial)/(\partial x)(xsin(8/x)) is sin(8/x)-(8cos(8/x))/x

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(\partial)/(\partial x)(xsin(8/x))

\frac{\partial}{\partial x} (x sin (\frac{8}{x}))

sin (\frac{8}{x}) - \frac{8 cos ( \frac{8}{x} )}{x}

Solution steps

\frac{\partial}{\partial x} (x sin (\frac{8}{x}))

Apply the Product Rule:

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

f = x, g = sin (\frac{8}{x})

= \frac{\partial}{\partial x} (x) sin (\frac{8}{x}) + \frac{\partial}{\partial x} (sin (\frac{8}{x})) x

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

\frac{\partial}{\partial x} (sin (\frac{8}{x})) = - \frac{8 cos ( \frac{8}{x} )}{x ^{2}}

= 1 \cdot sin (\frac{8}{x}) + (- \frac{8 cos ( \frac{8}{x} )}{x ^{2}}) x

Simplify

1 \cdot sin (\frac{8}{x}) + (- \frac{8 cos ( \frac{8}{x} )}{x ^{2}}) x : sin (\frac{8}{x}) - \frac{8 cos ( \frac{8}{x} )}{x}

= sin (\frac{8}{x}) - \frac{8 cos ( \frac{8}{x} )}{x}

x \to 0 lim (cos^{2} (x) \cdot x)

n = 0 \sum \infty \frac{1}{n + 16}

\frac{d ^{2} y}{d x ^{2}} + a \cdot y = 0

x \to 4 - lim (\frac{x - 4}{x - 3})

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

What is (\partial)/(\partial x)(xsin(8/x)) ?
The answer to (\partial)/(\partial x)(xsin(8/x)) is sin(8/x)-(8cos(8/x))/x