SUCCESSIVE DIFFERENTIATIONSuccessive 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 differentiatinga 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 RULEThis 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.Chain Rule Formula and ProofThere are two forms of chain rule formula as shown below.Chain Rule Formula 1:d/dx ( f(g(x) ) = f' (g(x)) · g' (x)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.Then by the chain rule formula,d/dx (sin 2x) = cos 2x · 2 = 2 cos 2xChain Rule Formula 2:We can assume the expression that is replacing "x" with "u" and applying the chain rule formula.dy/dx = dy/du · du/dxExample : To find d/dx (sin 2x), assume that y = sin 2x and 2x = u. Then y = sin u.By the chain rule formula,d/dx (sin 2x) = d/du (sin u) · d/dx(2x) = cos u · 2 = 2 cos u = 2 cos 2x. Double Chain RuleThere 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 isa 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/dxExample 1: y = (1+ cos 2x)2y' = 2( 1+ cos 2x) . (-sin 2x). (2)= - 4(1+ cos 2x) . sin2xExample 2: y = sin (cos (x2))y' = cos(cos (x2)). -sin (x2)). 2x= -2x sin (x2) cos (cos x2)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 replacing x.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.Solution:y = ln √x.f(x) = y is a composition of the functions ln(x) and √x, and therefore we can differentiate it using the chain rule.Assume that u = √x. Then y = ln u.By the chain rule formula,dy/dx = dy/du · du/dxdy/dx = d/du (ln u) · d/dx (√x)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 =