Question

What is the derivative of `tan^-1 (3x^2 +1)`?

Answer 1

`(6x)/(9x^4 + 6x^2 + 2)`

Explanation:

differentiate using the `color(blue)" chain rule "`

`d/dx [ f(g(x)) ] = f'(g(x)) . g'(x)`

and the standard derivative : D`(tan^-1 x) = 1/(1 + x^2)`
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f(g(x)) `= tan^-1(3x^2 + 1) rArr f'(g(x)) = 1/(1 + (3x^2 + 1)^2)`

and g(x) `= 3x^2 + 1 rArr g'(x) = 6x`
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`rArr f'(g(x)). g'(x) = 1/(1 +9x^4+ 6x^2 + 1 ) .6x`

`= (6x)/(9x^4 +6x^2 + 2)`