What is the derivative of arctanh(sqrt(x)) ?

The derivative of arctanh(sqrt(x)) is 1/(2sqrt(x)(1-x))

What is the first derivative of arctanh(sqrt(x)) ?

The first derivative of arctanh(sqrt(x)) is 1/(2sqrt(x)(1-x))

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\frac{d}{d x} (arctanh (x))

\frac{1}{2 x ( 1 - x )}

Solution steps

\frac{d}{d x} (arctanh (x))

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{1}{3 - 2 x}, 1

f (x) = x^{2} - 3 x

\frac{1}{3 + 2 x}

\frac{1}{1 - x}, 0

x \to 0 + lim (ln^{2} (x))

What is the derivative of arctanh(sqrt(x)) ?
The derivative of arctanh(sqrt(x)) is 1/(2sqrt(x)(1-x))
What is the first derivative of arctanh(sqrt(x)) ?
The first derivative of arctanh(sqrt(x)) is 1/(2sqrt(x)(1-x))