Solve d/dx 1/arcsinx

Evaluate

\frac{d}{d x} (\frac{1}{arcsin ( x )})

Rewrite the expression in exponential form

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

Use the chain rule \frac{d}{d x} (f (g)) = \frac{d}{d g} (f (g)) \times \frac{d}{d x} (g) where the g = arcsin (x), to find the derivative

\frac{d}{d g} (g^{- 1}) \times \frac{d}{d x} (arcsin (x))

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

- g^{- 2} \times \frac{d}{d x} (arcsin (x))

Use \frac{d}{d x} (arcsin x) = \frac{1}{1 - x ^{2}} to find derivative

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

Substitute back

- arcsin^{- 2} (x) \times \frac{1}{1 - x ^{2}}

Express with a positive exponent using a^{- n} = \frac{1}{a ^{n}}

- \frac{1}{arcsin ^{2} ( x )} \times \frac{1}{1 - x ^{2}}

Multiply the terms

- \frac{1}{arcsin ^{2} ( x ) \times 1 - x ^{2}}

Multiply by the Conjugate

- \frac{1 \times 1 - x ^{2}}{arcsin ^{2} ( x ) \times 1 - x ^{2} \times 1 - x ^{2}}

Calculate

- \frac{1 \times 1 - x ^{2}}{arcsin ^{2} ( x ) ( 1 - x ^{2} )}

Solution

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

Evaluate the derivative