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)#