If y = ⁻¹ ( 4 x 1+5 x^2 )+ ⁻¹ ( 3-2 x 2+3 x ) , then dy d x is equal to

Step-by-step solution

Let y = ⁻¹ ( 4x 1+5x^2 ) + ⁻¹ ( 3-2x 2+3x ) . The first term simplifies by observing that 4x 1+5x^2 matches the identity (A-B) = A-B 1+AB with A=5x and B=x , yielding ⁻¹(5x) - ⁻¹(x) . For the second term, ⁻¹ ( 3-2x 2+3x ) = ⁻¹ ( 2+3x 3-2x ) using the identity ⁻¹u = ⁻¹(1/u) . Dividing numerator and denominator by 3 gives ⁻¹ ( 2 3 +x 1- 2 3 x ) which matches the form ⁻¹ ( A+B 1-AB ) = ⁻¹A + ⁻¹B with A = 2 3 and B = x . Thus, ⁻¹ ( 3-2x 2+3x ) = ⁻¹ ( 2 3 ) + ⁻¹(x) . Substituting both simplified terms: y = [ ⁻¹(5x) - ⁻¹