What is the derivative of arctan(sqrt(x^2-1)) ?

The derivative of arctan(sqrt(x^2-1)) is 1/(xsqrt(x^2-1))

What is the first derivative of arctan(sqrt(x^2-1)) ?

The first derivative of arctan(sqrt(x^2-1)) is 1/(xsqrt(x^2-1))

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

\frac{1}{x x ^{2} - 1}

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{x x ^{2} - 1}

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