If x= e ^ ⁻¹ ( y -x^2 x^2 ) , then ( dy d x ) at x=1 is

Step-by-step solution

Begin with the given equation: x = e^ ⁻¹ ( y - x^2 x^2 ) . Take the natural logarithm of both sides to obtain x = ⁻¹ ( y - x^2 x^2 ) . Apply tangent to both sides, yielding ( x) = y - x^2 x^2 . Solve for y to express it as y = x^2 + x^2 ( x) = x^2(1 + ( x)) . Differentiate y with respect to x using the product rule. Let u = x^2 and v = 1 + ( x) , so u' = 2x and v' = 1 x ^2( x) . The derivative is dy dx = u'v + uv' = 2x(1 + ( x)) + x^2 1 x ^2( x) = 2x(1 + ( x)) + x ^2( x) . Evaluate at x = 1 , noting (1) = 0 , (0) =