What is the derivative of $sqrt(4+x^2)$ ?

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Answer 1 · 2016-04-02T15:09:36

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

Change the function $f(x) = sqrt(4 + x^2)$ into something more manageable by the laws of indices: $sqrt(u) = u^(1/2)$

$f(x) = (4 + x^2)^(1/2)$

This is approachable with the chain rule, which states that if $f(x) = g(h(x))$ then $f'(x) = h'(x)g'(h)$ , where $h$ is just shorthand for $h(x)$ .

With $h(x) = 4 + x^2$ and $g(x) = h^2(x)$ and simple differentiation,

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

Multiplying these two together as $h'(x)g'(h)$ ,

$(2x)/(2sqrt(4 + x^2)) = x/sqrt(4+x^2)$

What is the derivative of sqrt(4+x^2) sqrt(4+x^2) ?