Given that the tangent equation of the image of the function f (x) = (ax-6) / (x ^ 2 + b) at the point m (- 1, f (- 1)) is x + 2Y + 5 = 0, find the analytic expression graph of the function y = f (x) First, we substitute the point m x = - 1 into the tangent equation and get y = - 2 So the point (- 1, - 2) is the tangent point of F (x) If f (x) is derived, f '(x) = (- ax ^ 2 + 12x + AB) / (x ^ 2 + b) & # 178; So f '(- 1) = (- A-12 + AB) / (1 + b) & # 178; = - 1 / 2 (slope of tangent) ① And f (- 1) = (- a-6) / (B + 1) = - 2 That is, a = 2b-4 From (1) to (2) A = - 6, B = - 1 or a = 2, B = 3 And because (x ^ 2 + b) is the denominator, it is not zero, so B = - 1 is omitted a=2 b=3 Where f '(- 1) = (- A-12 + AB) / (1 + b) &# 178; = - 1 / 2 is based on what? Why is the slope obtained by derivation | Math enthusiast | Mathematical art

Given that the tangent equation of the image of the function f (x) = (ax-6) / (x ^ 2 + b) at the point m (- 1, f (- 1)) is x + 2Y + 5 = 0, find the analytic expression graph of the function y = f (x) First, we substitute the point m x = - 1 into the tangent equation and get y = - 2 So the point (- 1, - 2) is the tangent point of F (x) If f (x) is derived, f '(x) = (- ax ^ 2 + 12x + AB) / (x ^ 2 + b) & # 178; So f '(- 1) = (- A-12 + AB) / (1 + b) & # 178; = - 1 / 2 (slope of tangent) ① And f (- 1) = (- a-6) / (B + 1) = - 2 That is, a = 2b-4 From (1) to (2) A = - 6, B = - 1 or a = 2, B = 3 And because (x ^ 2 + b) is the denominator, it is not zero, so B = - 1 is omitted a=2 b=3 Where f '(- 1) = (- A-12 + AB) / (1 + b) &# 178; = - 1 / 2 is based on what? Why is the slope obtained by derivation

The physical meaning of derivative is tangent slope

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