Question

Find the derivative of y = x² / √(4 - x²).

a) 2x / (4 - x²)^(3/2)
b) 2x / (4 - x²)^(1/2)
c) (4 - x²) / (2x)
d) -2x / (4 - x²)^(3/2)

Answer 1

The correct derivative is a)
`\( (2x)/((4 - x^2)^(3/2)) \)`.

Step-by-step explanation:

The given function is
`\( y = (x^2)/(√(4 - x^2)) \)`, and we can find its derivative using the

quotient rule, which states that for a function
`\( f(x) = (g(x))/(h(x)) \)`, the derivative

`\( f'(x) \)`is given by
`\((g'(x)h(x) - g(x)h'(x))/((h(x))^2)\)`.

In this case, let
`\( g(x) = x^2 \)`and
`\( h(x) = √(4 - x^2) \)`. We find
`\( g'(x) = 2x \)` and

`\( h'(x) = -(x)/(√(4 - x^2)) \)`. Applying the quotient rule, the derivative
`\( y' \)` is given by:

`\[ y' = (2x \cdot √(4 - x^2) - x^2 \cdot \left(-(x)/(√(4 - x^2))\right))/((4 - x^2)) \]`

Simplifying further, we get:

`\[ y' = (2x(4 - x^2) + x^2 \cdot x)/((4 - x^2)^(3/2)) \]`

Combining like terms:

`\[ y' = (8x - 2x^3 + x^3)/((4 - x^2)^(3/2)) \]`

Simplifying the numerator:

`\[ y' = (-2x^3 + 9x)/((4 - x^2)^(3/2)) \]`

Factoring out ( -x ):

`\[ y' = (-x(2x^2 - 9))/((4 - x^2)^(3/2)) \]`

Finally, rearranging:

`\[ y' = (2x)/((4 - x^2)^(3/2)) \]`

So, the correct derivative is option (a).