What is the derivative of x^2tanh(1/x) ?

The derivative of x^2tanh(1/x) is 2xtanh(1/x)-sech^2(1/x)

What is the first derivative of x^2tanh(1/x) ?

The first derivative of x^2tanh(1/x) is 2xtanh(1/x)-sech^2(1/x)

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\frac{d}{d x} (x^{2} tanh (\frac{1}{x}))

2 x tanh (\frac{1}{x}) - sech^{2} (\frac{1}{x})

Solution steps

\frac{d}{d x} (x^{2} tanh (\frac{1}{x}))

Apply the Product Rule:

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

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

= \frac{d}{d x} (x^{2}) tanh (\frac{1}{x}) + \frac{d}{d x} (tanh (\frac{1}{x})) x^{2}

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

\frac{d}{d x} (tanh (\frac{1}{x})) = - \frac{sech ^{2} ( \frac{1}{x} )}{x ^{2}}

= 2 x tanh (\frac{1}{x}) + (- \frac{sech ^{2} ( \frac{1}{x} )}{x ^{2}}) x^{2}

Simplify

2 x tanh (\frac{1}{x}) + (- \frac{sech ^{2} ( \frac{1}{x} )}{x ^{2}}) x^{2} : 2 x tanh (\frac{1}{x}) - sech^{2} (\frac{1}{x})

= 2 x tanh (\frac{1}{x}) - sech^{2} (\frac{1}{x})

What is the derivative of x^2tanh(1/x) ?
The derivative of x^2tanh(1/x) is 2xtanh(1/x)-sech^2(1/x)
What is the first derivative of x^2tanh(1/x) ?
The first derivative of x^2tanh(1/x) is 2xtanh(1/x)-sech^2(1/x)