What is the derivative of xsin(pi/x) ?

The derivative of xsin(pi/x) is sin(pi/x)-(picos(pi/x))/x

What is the first derivative of xsin(pi/x) ?

The first derivative of xsin(pi/x) is sin(pi/x)-(picos(pi/x))/x

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\frac{d}{d x} (x sin (\frac{π}{x}))

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

Solution steps

\frac{d}{d x} (x sin (\frac{π}{x}))

Apply the Product Rule:

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

f = x, g = sin (\frac{π}{x})

= \frac{d x}{d x} sin (\frac{π}{x}) + \frac{d}{d x} (sin (\frac{π}{x})) x

\frac{d x}{d x} = 1

\frac{d}{d x} (sin (\frac{π}{x})) = - \frac{π cos ( \frac{π}{x} )}{x ^{2}}

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

Simplify

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

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

f (x) = tan (ln (x))

\int_{- 3}^{5} (2 y - (y^{2} - 15)) d y

2 x^{\frac{3}{2}}

y^{^{'}} = \frac{( y - y ^{2} ) t}{1 + t ^{2}}

\int 2 x sin (2 x) d x

What is the derivative of xsin(pi/x) ?
The derivative of xsin(pi/x) is sin(pi/x)-(picos(pi/x))/x
What is the first derivative of xsin(pi/x) ?
The first derivative of xsin(pi/x) is sin(pi/x)-(picos(pi/x))/x