Consider the following for the two (02) items that follow: Let y = ⁻¹ (x - 4x^3 27 ) . What is dy dx equal to?

Step-by-step solution

Let x = 3 . Then = ⁻¹ ( x 3 ) . Substituting x = 3 into the given expression: y = ⁻¹ (3 - 4(3 )^3 27 ) y = ⁻¹ (3 - 4(27 ^3 ) 27 ) y = ⁻¹(3 - 4 ^3 ) Using the trigonometric identity (3 ) = 3 - 4 ^3 : y = ⁻¹( 3 ) Assuming principal values, we get: y = 3 y = 3 ⁻¹ ( x 3 ) Differentiating with respect to x using the chain rule: dy dx = 3 1 1 - ( x 3 )^2 d dx ( x 3 ) dy dx = 3 1 1 - x^2 9 1 3 dy dx = 1 9 - x^2 9 dy dx = 3 9 - x^2 Answer: 3 9-x^2