SUCCESSIVE DIFFERENTIATION Successive differentiation The higher order differential coefficients are of utmost importance in scientific and engineering applications Let y f x be a function of x Then the result of differentiating y with respect to x is defined as the derivative or the first derivative of y with respect to x and it is denoted by frac dy dx Then the result of differentiating f x with respect to x is defined as the second derivative of the function and it is denoted as frac d 2 y d x 2 This process can be defined to n times known as successive differentiation Successive differentiation is the process of differentiating a given function several times and the results are called successive derivatives Standard Differentiation Formula i d dx c 0 where cc is constant ii d dx x 1 iii d dx cx c where cc is constant iv d dx x n nx n 1 power rule v d dx f x n n f x n 1f x vi d dx f x g x d dx f x d dx g x sum rule vii d dx f x g x d dx f x d dx g x difference rule viii d dx f x g x f x d dx g x g x d dx f x product rule ix d dx f x g x g x d dxf x f x d dxg x g x 2 Quotient rule CHAIN RULE This chain rule is also known as the outside inside rule or the composite function rule or function of a function rule It is used only to find the derivatives of the compiste functions The Theorem of Chain Rule Let f be a real valued function that is a composite of two functions g and h i e f g o h Suppose u h x where du dx and dg du exist then this could be expressed as change in f change in x change in g change in u change in u change in x This is given as Leibniz notation in the form of an equation as df dx dg du du dx Example To find the derivative of d dx sin 2x express sin 2x f g x where f x sin x and g x 2x Chain Rule Formula and Proof There are two forms of chain rule formula as shown below Chain Rule Formula 1 d dx f g x f g x g x Then by the chain rule formula d dx sin 2x cos 2x 2 2 cos 2x Chain Rule Formula 2 dy dx dy du du dx By the chain rule formula We can assume the expression that is replacing x with u and applying the chain rule formula Example To find d dx sin 2x assume that y sin 2x and 2x u Then y sin u d dx sin 2x d du sin u d dx 2x cos u 2 2 cos u 2 cos 2x Double Chain Rule There could be nested functions one over the other where the functions depend on more than one variable The chain of smaller derivatives is multiplied together to get the overall derivative Let there be 3 functions u v w A function f is a composite of u v and w The chain rule is extended here If a function is a composition of 3 functions we apply the chain rule twice When f u o v o w df dx df du du dv dv dw dw dx Example 1 y 1 cos 2x 2 y 2 1 cos 2x sin 2x 2 4 1 cos 2x sin2x Example 2 y sin cos x2 y cos cos x2 sin x2 2x 2x sin x2 cos cos x2 replacing x Solution y ln x rule Assume that u x Then y ln u By the chain rule formula dy dx dy du du dx dy dx d du ln u d dx x Note We do not need to remember the chain rule formula Instead we can just apply the derivative formulas which are in terms of x and then multiply the result by the derivative of the expression that is For example d dx x2 1 3 3 x2 1 2 d dx x2 1 3 x2 1 2 2x 6x x2 1 2 Example 1 Find the derivative of y ln x using the chain rule f x y is a composition of the functions ln x and x and therefore we can differentiate it using the chain dy dx 1 u 1 2 x dy dx 1 x 1 2 x dy dx 1 2x because u 1 2 x y cos 2x2 1 Answer dy dx 1 2x