What is the derivative of arcsin(x/(sqrt(1+x^2))) ?

The derivative of arcsin(x/(sqrt(1+x^2))) is 1/(1+x^2)

What is the first derivative of arcsin(x/(sqrt(1+x^2))) ?

The first derivative of arcsin(x/(sqrt(1+x^2))) is 1/(1+x^2)

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is the derivative of arcsin(x/(sqrt(1+x^2))) ?
The derivative of arcsin(x/(sqrt(1+x^2))) is 1/(1+x^2)
What is the first derivative of arcsin(x/(sqrt(1+x^2))) ?
The first derivative of arcsin(x/(sqrt(1+x^2))) is 1/(1+x^2)