MTH 114 Final: MTH 114
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Second derivative: d2y/dx2 = d/dx( dy/dx) = d/dt (dy/dx)/ dx/dt. Since f inverse is reflection over line y=x which gives f: f"(x) at (x,y) is equal to f inverse (x) = 1/f"(x) at (y,x) Confirmed using implicit: y=f^-1(x) is same as x =f(y):1=d/dx x = d/dx [f(y)] = f"(y) dy/dx. Dy/dx = 1/f prime of y = 1/f prime of f inverse of x. 6. 3. 1 theorem: suppose domain of function is open interval on which f"(x) greater than 0 or on which f prime of x less than 0. Then f is one-to-one, f prime of x is diff at all values of x in range, and derivative of f inverse prime of is formula above. Sin inverse x + cosines inverse x = pi/2. Tan of sin inverse of x = x/(1-x^2)^0. 5. Sin(sec inverse of x)= (x^2-1)^0. 5/x so that x is greater than of equal to one.
