What is the derivative of $x * ((4-x^2)^(1/2))$ ?

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Answer 1 · 2016-10-24T08:03:24

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

$y=x*sqrt(4-x^2)$

I'll differentiate using the product and chain rules.

Product rule
$y=f(x)*g(x)$

$y'=f'(x)g(x) +g'(x)f(x)$

$y=f(g(x))$

$y' = f'(g(x))*g'(x)$

Differentiate each part
$f(x)=x$

$f'(x)=1$

$g(x)=sqrt(4-x^2)$

( $y=f(g(x)) => y' = f'(g(x))*g'(x)$ )

$g'(x)=1/2(4-x^-2)^(-1/2)(-2x)$

so using the product rule

$y'=f'(x)g(x) +g'(x)f(x)$

$y'=sqrt(4-x^2) +(1/2(4-x^-2)^(-1/2)(-2x))(x)$

$y'=sqrt(4-x^2) +(-x^2)/sqrt(4-x^2)$

$y'=(4-x^2 -x^2)/sqrt(4-x^2)$

$y'=(-2x^2+4)/(sqrt(-x^2+4))$

which is the derivative of the function.

What is the derivative of x * ((4-x^2)^(1/2)) x * ((4-x^2)^(1/2))?