What is the derivative of arcsinh(tan(x)) ?

The derivative of arcsinh(tan(x)) is sqrt(sec^2(x))

What is the first derivative of arcsinh(tan(x)) ?

The first derivative of arcsinh(tan(x)) is sqrt(sec^2(x))

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\frac{d}{d x} (arcsinh (tan (x)))

sec^{2} (x)

Solution steps

\frac{d}{d x} (arcsinh (tan (x)))

Apply the chain rule:

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

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

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

= \frac{1}{( tan ( x ) ) ^{2} + 1} sec^{2} (x)

Simplify

\frac{1}{tan ^{2} ( x ) + 1} sec^{2} (x) : sec^{2} (x)

= sec^{2} (x)

What is the derivative of arcsinh(tan(x)) ?
The derivative of arcsinh(tan(x)) is sqrt(sec^2(x))
What is the first derivative of arcsinh(tan(x)) ?
The first derivative of arcsinh(tan(x)) is sqrt(sec^2(x))