If y = ⁻¹ [ 4 2 x 2 x-6 ^2 x ] , then dy d x at x=0 is

Step-by-step solution

Given y = ⁻¹ [ 4 2x 2x - 6 ^2 x ] , use the identities 2x = 2 x x and 2x = ^2 x - ^2 x to simplify the inner expression. y = ⁻¹ [ 8 x x ^2 x - 7 ^2 x ] Dividing numerator and denominator by ^2 x (where x 0 ) gives: y = ⁻¹ [ 8 x 1 - 7 ^2 x ] Recognize the form ⁻¹ ( A+B 1-AB ) = ⁻¹ A + ⁻¹ B with A = 7 x and B = x . y = ⁻¹(7 x) + ⁻¹( x) = ⁻¹(7 x) + x Differentiate using the chain rule: dy dx = 1 1+(7 x)^2 7 ^2 x + 1 = 7 ^2 x 1+49 ^2 x + 1 At x=0 , 0 = 0 and 0 = 1 , so: . dy dx |_ x=0 = 7 1^2 1 + 49 0^2 + 1 = 7 + 1 = 8