If y = ⁻¹ ( ( e x^3 ) ex^3 ) + ⁻¹ ( (e^4x^3) ( e x¹² ) ) , x (e^ - 1 3 , e^ 1 12 ) then dy dx is equal to...

Step-by-step solution

Given function is: y = ⁻¹ ( ( e x^3 ) (ex^3) ) + ⁻¹ ( (e^4x^3) ( e x¹² ) ) Using the properties of logarithms, we can simplify the arguments: ( e x^3 ) (ex^3) = e - x^3 e + x^3 = 1 - 3 x 1 + 3 x (e^4x^3) ( e x¹² ) = e^4 + x^3 e - x¹² = 4 + 3 x 1 - 12 x Substituting these back into the expression for y : y = ⁻¹ ( 1 - 3 x 1 + 1 3 x ) + ⁻¹ ( 4 + 3 x 1 - 4 3 x ) We use the inverse trigonometric identities: ⁻¹ ( a - b 1 + ab ) = ⁻¹a - ⁻¹b (valid for ab > -1 ) ⁻¹ ( a + b 1 - ab ) = ⁻¹a + ⁻¹b (valid for ab For the first t