Question

How do you differentiate `y = x * sqrt (4 - x^2)`?

Answer 1

`dy/dx =( 2 (2 - x^2 ))/sqrt(4 - x^2 )`

Explanation:

differentiate using the 'product rule' and 'chain rule':

`dy/dx = x . d/dx (4 - x^2)^(1/2 )+ (4 - x^2)^(1/2) .d/dx (x)`

`dy/dx = x(1/2 (4 - x^2)^(-1/2) .d/dx (4 - x^2 ) )+ (4 - x^2 )^(1/2) . 1`

`dy/dx = x ( 1/2 (4 - x^2 )^(-1/2) .(-2x)) + (4 -x^2)^(1/2)`

`dy/dx = - x^2 (4 - x^2)^(-1/2) + (4 - x^2 )^(1/2)`

[ common factor of `(4 - x^2 )^(-1/2) ]`

`dy/dx = (4 - x^2 )^(-1/2) [ -x^2 + 4 - x^2]`

`dy/dx = (4 - x^2 )^(-1/2) . (4 - 2x^2)`

`rArr dy/dx =( 2(2 - x^2))/sqrt(4 - x^2)`