The Chain Rule Academic Resource Center In This Presentation… • We will give a definition • Prove the chain rule • Learn how to use it • Do example problems Definition • In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f ∘ g in terms of the derivatives of f and g. Definition • If g is differentiable at x and f is differentiable at g(x), then the composite function F = f ∘ g defined by F(x) = f(g(x)) is differentiable at x and F’ is given by the product F’(x) = f’(g(x)) · g’(x) • In Leibniz notation, if y = f (u) and u = g(x) are both differentiable functions, then 푑푦 푑푦 푑푢 = 푑푥 푑푢 푑푥 Definition • The Chain Rule can be written either in the prime notation (f ∘ g)’(x) = f’(g(x))·g’(x) • Or, if y = f(u) and u = g(x), in Leibniz notation: 푑푦 푑푦 푑푢 = 푑푥 푑푢 푑푥 Proof of the Chain Rule • Recall that if y = f(x) and x changes from a to a + Δx, we defined the increment of y as Δy = f(a + Δx) – f(a) • According to the definition of a derivative, we have Δy lim = f’(a) Δx→0 Δx Proof of the Chain Rule • So if we denote by ε the difference between the difference quotient and the derivative, we obtain Δy lim 휀 = lim ( - f’(a)) Δx→0 Δx→0 Δx = f’(a) – f’(a) = 0 • But Δy ε = - f’(a) Δy = f’(a) Δx + ε Δx Δx Proof of the Chain Rule • If we define ε to be 0 when Δx = 0, the ε becomes a continuous function of Δx. Thus, for a differentiable function f, we can write Δy = f’(a) Δx + ε Δx, where ε 0 as x 0 (1) • and ε is a continuous function of Δx. This property of differentiable functions is what enables us to prove the Chain Rule. Proof of the Chain Rule • Suppose u = g(x) is differentiable at a and y = f(u) is differentiable at b = g(a). If Δx is an increment in x and Δu and Δy are the corresponding increment in u and y, then we can use Equation (1) to write Δu = g’(a) Δx + ε1 Δx = * g’(a) + ε1 ]Δx (2) where ε1 0 as Δx 0. Proof of the Chain Rule • Similarly Δy = f’(b)Δu + ε2Δu = *f’(b) + ε2]Δu (3) where ε2 0 as Δu 0 • If we now substitute the expression for Δu from equation (2) into equation (3), we get Δy = * f’(b) + ε2 +* g’(a) + ε1 ]Δx Δ푦 • So = * f’(b) + ε2 +* g’(a) + ε1 ] Δ푥 Proof of the Chain Rule • As Δx 0, Equation (2) shows that Δu 0. So both ε1 0 and ε2 0 as Δx 0. • Therefore 푑푦 Δ푦 = lim 푑푥 Δx→0 Δ푥 = lim * f’(b) + ε2 +* g’(a) + ε1 + Δx→0 = f’(b)g’(a) = f’(g(a))g’(a) • This proves the Chain Rule. How to use the Chain Rule • In using the Chain Rule, we work from the outside to the inside. First, we differentiate the outer function f [ at the inner function g(x) ] and then we multiply by the derivative of the inner function. 푑 f (g(x)) = f’ (g(x)) · g’(x) 푑푥 Outer Evaluated Derivative Evaluated Derivative function at inner of outer at inner of inner function function function function Examples • Differentiate y = sin ( x2). As we can see, the outer function is the sine function and the inner function is the squaring function, so the Chain Rule gives 푑푦 푑 2 2 = sin ( x ) = cos ( x ) · 2x 푑푥 푑푥 2 = 2xcos( x ) Outer Evaluated Derivative Evaluated Derivative function at inner of outer at inner of inner function function function function The power rule combined with the Chain Rule • This is a special case of the Chain Rule, where the outer function f is a power function. If y = *g(x)+푛, then we can write y = f(u) = u푛 where u = g(x). By using the Chain Rule an then the Power Rule, we get 푑푦 푑푦 푑푢 푛;1 푑푢 푛;1 = = nu = n*g(x)+ g’(x) 푑푥 푑푢 푑푥 푑푥 Examples 100 • Differentiate y = (x3 − 1) . 3 Taking u = g(x) = x − 1 and n = 100, we have 푑푦 푑 3 100 = (x − 1) 푑푥 푑푥 3 99 푑 3 = 100(x − 1) (x − 1) 푑푥 3 99 2 = 100(x − 1) · 3x 99 = 300 x2(x3 − 1) Examples • Find the derivative of the function 푡 ;2 g(t) = ( )9 2푡:1 Combining the Power Rule, Chain Rule, and Quotient Rule, we get 푡 ;2 8 푑 푡;2 g’(t) = 9 ( ) 2푡:1 푑푡 2푡:1 푡 ;2 8 2푡:1 ∙1;2(푡;2) 45(푡;2)8 = 9 = 2푡:1 (2푡:1)2 (2푡:1)10 Examples • Differentiate y = (2푥 + 1)5(푥3−푥 + 1)4. In this example we must use the Product Rule before using the Chain Rule: 푑푦 5 푑 3 4 3 4 푑 5 = (2푥 + 1) (푥 −푥 + 1) + (푥 −푥 + 1) (2푥 + 1) 푑푥 푑푥 푑푥 푑 = (2푥 + 1)5· 4(푥3−푥 + 1)3 (푥3−푥 + 1) + 푑푥 푑 5(2푥 + 1)4(푥3−푥 + 1)4 (2x+1) 푑푥 Examples = 4(2푥 + 1)5(푥3−푥 + 1)3(3푥2-1) + 5(2푥 + 1)4(푥3−푥 + 1)4·2 = 25(2푥 + 1)4(푥3−푥 + 1)3 (17푥3+6푥2-9x+3) More Examples • The reason for the name “Chain Rule” becomes clear when we make a longer chain by adding another link. Suppose that y = f(u), u = g(x), and x = h(t), where f, g, and h are differentiable functions. Then, to compute the derivative of y with respect to t, we use the Chain Rule twice: 푑푦 푑푦 푑푥 푑푦 푑푢 푑푥 = = 푑푡 푑푥 푑푡 푑푢 푑푥 푑푡 More Examples • If f(x) = sin(cos(tanx)), then 푑 f’(x) = cos(cos(tanx)) cos(tanx) 푑푥 푑 = cos(cos(tanx))[-sin(tanx)] (tanx) 푑푥 = - cos(cos(tanx))sin(tanx)sec2x More Examples • Differentiate y = sec x3. 푑푦 1 푑 3 = (sec x ) 푑푥 2 sec x3 푑푥 1 3 3 푑 3 = sec x tan x (x ) 2 sec x3 푑푥 3x2sec x3tan x3 = 2 sec x3 Thanks! References • Calculus – Stewart 6th Edition • Section 2.5 “The Chain Rule” • Appendixes A1, F “Proofs of Theorems .