What is the derivative of 2sqrt(x)tanh(sqrt(x)) ?

The derivative of 2sqrt(x)tanh(sqrt(x)) is (tanh(sqrt(x)))/(sqrt(x))+sech^2(sqrt(x))

What is the first derivative of 2sqrt(x)tanh(sqrt(x)) ?

The first derivative of 2sqrt(x)tanh(sqrt(x)) is (tanh(sqrt(x)))/(sqrt(x))+sech^2(sqrt(x))

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

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

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

f = x, g = tanh (x)

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

\frac{d}{d x} (x) = \frac{1}{2 x}

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

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

Simplify

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

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

- \frac{9}{( 6 t + 8 ) ^{\frac{3}{2}}}

x \to 1 - lim (\frac{4}{x ^{3} - 1})

\frac{d}{d x} (3 (x)^{3} + cos (x))

x \to 0 lim (x^{2} csc^{2} (x))

\frac{d}{d x} (tan (x) - x + 1)

What is the derivative of 2sqrt(x)tanh(sqrt(x)) ?
The derivative of 2sqrt(x)tanh(sqrt(x)) is (tanh(sqrt(x)))/(sqrt(x))+sech^2(sqrt(x))
What is the first derivative of 2sqrt(x)tanh(sqrt(x)) ?
The first derivative of 2sqrt(x)tanh(sqrt(x)) is (tanh(sqrt(x)))/(sqrt(x))+sech^2(sqrt(x))