Find the rst, second and third derivative of the given function 7. x): _4 (H2)2 2:54 PM @ 65% cdn.fbsbx.com Done MATH 401 - DIFFERENTIAL CALCULUS 2.4 Higher-Order Derivatives The derivative f' of a function f is itself a function and hence may have a derivative of its own. If f' is differentiable, then its derivative is denoted by f" and is called the second derivative of f . As long as we have differentiability, we can continue the process of differentiating to obtain the third, fourth, fifth, and even higher derivatives of f. These successive derivatives are denoted by f f " = ( 1), f " = (1"), f* =(f"),f =(f' )'.... Using the same functional notation y= f(x), we will have the following symbols for higher derivatives: First Derivative: "' . f ( x ) , dy d() . [f(x)] . D.f (x) Second Derivative: y" , f"(x) , dy d' dx' ' dp () . 7 [/ ( x ) ] . D. S (x ) Third Derivative: y" , f" (x ) , do [f (x ) ] . D'S ( x ) "" Derivative: v . f"(x) , do . de() . +.[(x)] . D.f(x) The symbols dy d'y dy are called Leibniz notations. dx ' dx ' dx Example1. Given f(x)=x' -2x -3x +5, find f'(x), f"(x) and f"(x) Solution: f(x)= 3x' -4x-3 "(x)= 6x -4 f" (x)=6 Example2. Find f'(x), f"(x) and f"(x) given that f(x)= x3 + 2x-' Solution: f (x) = 2.x +2(-x-?)=2x-2x-2