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

The derivative of arctanh(sin(x)) is sec(x)

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

The first derivative of arctanh(sin(x)) is sec(x)

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

sec (x)

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (sin (x)) = cos (x)

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

Simplify

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

= sec (x)

What is the derivative of arctanh(sin(x)) ?
The derivative of arctanh(sin(x)) is sec(x)
What is the first derivative of arctanh(sin(x)) ?
The first derivative of arctanh(sin(x)) is sec(x)