How do you find the derivative of the function: $(arctan(6x^2 +5))^2$ ?

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Answer 1 · 2016-01-23T15:59:31

$(24xarctan(6x^2+5))/((6x^2+5)^2+1)$

This will require a little chain rule.

The first issue is the second power, which can be dealt with as such: $d/dx[u^2]=2u*u'$ . We know that $u=arctan(6x^2+5)$ , so

$f'(x)=2arctan(6x^2+5)d/dx[arctan(6x^2+5)]$

Now, we must deal with the $arctan$ function, which uses the rule $d/dx[arctan(u)]=(u')/(u^2+1)$ , and we have $u=6x^2+5$ , giving us

$f'(x)=2arctan(6x^2+5)*(d/dx[6x^2+5])/((6x^2+5)^2+1)$

Simplify:

$f'(x)=(2arctan(6x^2+5)*12x)/((6x^2+5)^2+1)$

$f'(x)=(24xarctan(6x^2+5))/((6x^2+5)^2+1)$

How do you find the derivative of the function: (arctan(6x^2 +5))^2(arctan(6x^2 +5))^2?