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

The derivative of xsin(y/x) is sin(y/x)+(cos(y/x)(x(dy)/(dx)-y))/x

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

The first derivative of xsin(y/x) is sin(y/x)+(cos(y/x)(x(dy)/(dx)-y))/x

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\frac{d}{d x} (x sin (\frac{y}{x}))

sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x}

Solution steps

\frac{d}{d x} (x sin (\frac{y}{x}))

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

What is the derivative of xsin(y/x) ?
The derivative of xsin(y/x) is sin(y/x)+(cos(y/x)(x(dy)/(dx)-y))/x
What is the first derivative of xsin(y/x) ?
The first derivative of xsin(y/x) is sin(y/x)+(cos(y/x)(x(dy)/(dx)-y))/x