What's the derivative of $arctan(x^3/3)$ ?

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Answer 1 · 2016-02-16T23:16:24

$(9x^2)/(x^6+9)$

We can use the chain rule, which states that

$d/dx(f(g(x))=f'(g(x))*g'(x)$

In the case of an arctangent function, it will help to know that

$d/dx(arctan(x))=1/(x^2+1)$

When we apply this to the chain rule, we see that

$d/dx(arctan(g(x)))=1/((g(x))^2+1)*g'(x)$

When differentiating $arctan(x^3/3)$ , we see that $g(x)=x^3/3$ , yielding a derivative of:

$d/dx(arctan(x^3/3))=1/((x^3/3)^2+1)*d/dx(x^3/3)$

Through the power rule, we know that $d/dx(x^3/3)=x^2$ . The rest becomes simplification:

$=1/(x^6/9+1)*x^2$

$=x^2/((x^6+9)/9)$

$=(9x^2)/(x^6+9)$

What's the derivative of arctan(x^3/3)arctan(x^3/3)?