What is the derivative of y=xsin(1/x) ?

The derivative of y=xsin(1/x) is sin(1/x)-(cos(1/x))/x

What is the first derivative of y=xsin(1/x) ?

The first derivative of y=xsin(1/x) is sin(1/x)-(cos(1/x))/x

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derivative of

y = x sin (\frac{1}{x})

sin (\frac{1}{x}) - \frac{cos ( \frac{1}{x} )}{x}

Solution steps

\frac{d}{d x} (x sin (\frac{1}{x}))

Apply the Product Rule:

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

f = x, g = sin (\frac{1}{x})

= \frac{d x}{d x} sin (\frac{1}{x}) + \frac{d}{d x} (sin (\frac{1}{x})) x

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

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

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

Simplify

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

= sin (\frac{1}{x}) - \frac{cos ( \frac{1}{x} )}{x}

3 x^{2} (x^{3} + 1)^{7}

x \to 0 lim (\frac{sin ^{2} ( 4 x )}{x})

x \to 0 + lim (- \frac{1}{sin ( 2 x )})

4 (t + 3)

\frac{d}{d x} (y x^{y - 1})

What is the derivative of y=xsin(1/x) ?
The derivative of y=xsin(1/x) is sin(1/x)-(cos(1/x))/x
What is the first derivative of y=xsin(1/x) ?
The first derivative of y=xsin(1/x) is sin(1/x)-(cos(1/x))/x