How do you use the chain rule to differentiate $y=(x^2+1)^(1/2)$ ?

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Answer 1 · 2017-02-26T02:08:51

$dy/dx = x/sqrt(x^2+1)$

$y = (x^2+1)^(1/2)$

$dy/dx = 1/2*(x^2+1)^(-1/2) * d/dx (x^2+1)$

Apply the power rule:

$dy/dx= 1/2*(x^2+1)^(-1/2) * (2x+0)$

$= (cancel2x)/(cancel2* sqrt(x^2+1)$

$= x/sqrt(x^2+1)$

Answer 2 · 2017-02-26T05:21:17

Recall that the chain rule is similar as the power rule but has one more step.

The chain rule states that the derivative of $f(g(x))$ is $f'(g(x))*g'(x)$ .

In this instance,

$f(x) = g(x)^(1/2)$

$g(x) = x^2 + 1$

So, the derivative of the composition $f(g(x))$ is $f'(g(x))*g'(x)$ , which is:

$(1/2) * (x^2 + 1)^(-1/2) * 2x$

Simplified, we get:

$x / sqrt(x^2 + 1)$

How do you use the chain rule to differentiate y=(x^2+1)^(1/2)y=(x^2+1)^(1/2)?