Question

How to find the derivative of this function f(x)?
Please explain. Thanks

Answer 1

Use the quotient and chain rules.

Quotient rule:

`\left((x)/(√(1-x^2))\right)' = ((x)' √(1 - x^2) - x \left(√(1-x^2)\right)')/(\left(√(1-x^2)\right)^2)`

Chain rule:

`\left(√(1-x^2)\right)' = \frac1{2√(1-x^2)} (1-x^2)' = -\frac x{√(1-x^2)}`

`\implies \left((x)/(√(1-x^2))\right)' = (√(1 - x^2) + (x^2)/(√(1-x^2)))/(1-x^2)`

Now just simplify.

`\left((x)/(√(1-x^2))\right)' = (√(1 - x^2) + (x^2)/(√(1-x^2)))/(1 - x^2) \\\\ ~~~~~~~~~~~~ = \frac1{√(1-x^2)} + (x^2)/((1-x^2)^(3/2)) \\\\ ~~~~~~~~~~~~ = \frac1{√(1-x^2)} \left(1 + (x^2)/(1-x^2)\right) \\\\ ~~~~~~~~~~~~ = \frac1{√(1-x^2)} \cdot \frac1{1-x^2} \\\\ ~~~~~~~~~~~~ = \frac1{(1-x^2)^(3/2)}`