What is (\partial)/(\partial x)(x^2(1+sin(1/x))) ?

The answer to (\partial)/(\partial x)(x^2(1+sin(1/x))) is 2x(1+sin(1/x))-cos(1/x)

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(\partial)/(\partial x)(x^2(1+sin(1/x)))

\frac{\partial}{\partial x} (x^{2} (1 + sin (\frac{1}{x})))

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

Solution steps

\frac{\partial}{\partial x} (x^{2} (1 + sin (\frac{1}{x})))

Apply the Product Rule:

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

f = x^{2}, g = 1 + sin (\frac{1}{x})

= \frac{\partial}{\partial x} (x^{2}) (1 + sin (\frac{1}{x})) + \frac{\partial}{\partial x} (1 + sin (\frac{1}{x})) x^{2}

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

Simplify

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

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

x \to 2.5 lim (g (x))

\int \frac{1}{t ^{2} + 6 t + 25} d t

(3, 2), (7, - 4)

x^{2}, 2 x - x^{2}, 0, 1

\int x (x^{4} - 1)^{2} d x

What is (\partial)/(\partial x)(x^2(1+sin(1/x))) ?
The answer to (\partial)/(\partial x)(x^2(1+sin(1/x))) is 2x(1+sin(1/x))-cos(1/x)