Question

if for x€(0,1/4) the derivative of tan^-1((6x x^1/2.)/1-9x^3) is x^1/2 g(x) equals to ???

Answer 1

Hello Dear

y = tan-¹(6x✓x/1-9x³) and x€(0,1/4). { given

so
tan-¹[2(3x³/²)/ 1- ( 3x)³/²]. --------1

we know that 2tan-¹ x = tan-¹(2x/1-x)
use. it here

we get
= 2tan-¹(3x³/²)


so
dy/dx(2tan-¹(3x³/²)
we know. dy/dx(tan-¹(x)) => 1/1+x²
use it here !!

we get.
dy/dx. =. 2 ×. 1/1+9x³. × 3×3/2 × x½
=> 9/1+x³✓3
=> g(x) = 9/1+9x³
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