Section 3 4B The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x then dy dx = dy du du dx or If y = f (u) then f (u) u The Chain Rule with the Power Rule Theorem: If a and n are a real numbers If y = a u n then a n u n-1 u or in English If y = a ( some function of x inside the brackets ) n then a n ( some function of x inside the brackets ) n-1 the derivative of the function inside the brackets Example 1 The Chain Rule with the Power Rule Find y if y = ( 5x 2) 3 if we let u = 5x 2 then y = ( u) 3 where u = 5x 2 and u = 5 n(u) n 1 u n 3 u n 1 64 748 ( 5x 2) 2 u 5 15( 5x 2) 2 Math 400 3 4B Chain Rule Page 1 of 11 2013 Eitel
Example 2 The Chain Rule with the Power Rule Find y if y = 2( 3x 2 + 5x) 4 Think of the outside function as 2(u) 4 take the derivative of the outside function 2 4(u) 3 3 8 3x 2 + 5x now multiply that by the derivative of the inside function if u = 3x 2 + 5x then u = 6x + 5 a n 67 8 2 4 u n 1 647 48 ( 3x 2 + 5x) 3 64 7 u 48 ( 6x + 5) 8( 3x 2 + 5x) 3 ( 6x + 5) Example 3 The Chain Rule with the Power Rule Find y if y = 4( tan x) 3 Think of the outside function as 4(u) 3 take the derivative of the outside function 4 3(u) 2 12( tan x) 2 now multiply that by the derivative of the inside function if u = tan x then u = sec 2 x a n 64 74 8 64 7 u 48 12 ( tan x) 2 sec 2 x u n 1 12tan 2 x sec 2 x Math 400 3 4B Chain Rule Page 2 of 11 2013 Eitel
Example 4 The Chain Rule with the Power Rule Find y if y = 3 ( 6x 2 1) = ( 6x 2 1) 1/3 Think of the outside function as (u) 1/3 take the derivative of the outside function 1 3 (u) 2/3 2/3 1 3 6x2 1 now multiply that by the derivative of the inside function if u = 6x 2 1 then u = 12x n 1 3 u n 1 6 47 4 8 ( 6x 2 1) 2/3 67 u 8 ( 12x) 1 3 12x ( 6x2 1) 2/3 4x ( 6x 2 1) 2/3 Example 5 The Chain Rule with the y = Sin (u) Find f if f (x) = sin( 3x 4 ) if y = f (u) then f (u) u Think of the outside function as sin(u) take the derivative of the outside function to get cos(u) cos 3x 4 now multiply that by the derivative of the inside function if u = 3x 4 then u = 12x 3 64 f 7 (u) 48 cos 3x 4 u 12x 3 f (x) = 12x 3 cos 3x 4 Math 400 3 4B Chain Rule Page 3 of 11 2013 Eitel
Example 6 The Chain Rule with y = e u Find f (x) if f (x) = e x 2 if y = f (u) than f (u) u Think of the outside function as e u take the derivative of the outside function to get e u e x2 now multiply that by the derivative of u if u = x 2 then u = 2x f (u) e x 2 u 2x 2x e x 2 Example 7 Find f (x) if f (x) = e x = e x1/2 if y = f (u) than f (u) u Think of the outside function as e u take the derivative of the outside function to get e u e x1/2 now multiply that by the derivative of u if u = x 1/2 then u = 1 2 x 1/2 f (u) e x1/2 67 u 8 1 2 x 1/2 e x 2x 1/2 e x 2 x Math 400 3 4B Chain Rule Page 4 of 11 2013 Eitel
Example 8 The Chain Rule with y = ln( u) Find y if y = ln( 3x 4 + 2x 3 ) if u = 3x 4 + 2x 3 then u =12x 3 + 6x 2 if y = ln(u) then u u 12x 3 + 6x 2 3x 4 + 2x 3 6x 2 2x +1 x 3 3x + 2 6 2x +1 x 3x + 2 Example 9 The Chain Rule with y = ln( u) Find y if y = ln y = ln 3x 2 5x 3x 2 5x 1/2 = 1 2 ln ( 3x2 5x) if u = 3x 2 5x then u = 6x 5 if y = ln(u) then u u 1 2 6x 5 3x 2 5x 6x 5 2 3x 2 5x 6x 5 2x 3x 5 Math 400 3 4B Chain Rule Page 5 of 11 2013 Eitel
Example 10 The Chain Rule with y = ln Find y if y = ln u = ln u lnv v x 2 1 sin(2x) y = ln ( x 2 1) ln sin(2x) if y = ln(u) then u u der ofx 2 2x x 2 1 der 64 of 7 sin(2 48 x) cos(2x) 2 sin2x 2x x 2 1 2 cos(2x) sin(2x) Example 11 The Chain Rule with y = ln Find y if y = ln y = ln x sin(2x) x sin(2x) e x2 ln e x 2 u = ln u lnv v y = ln(x) + ln(sin(2x)) x 2 ln ( e) y = ln(x) + ln(sin(2x)) x 2 if y = ln(u) then u u der of x 1 + x der 64 of 7 sin(2 48 x) cos(2x) 2 sin(2x) der of x 2 2x 1 x + 2 cos(2x) sin(2x) 2x Math 400 3 4B Chain Rule Page 6 of 11 2013 Eitel
Repeated use of the Chain Rule then f g If y = f ( g [ ( h { x ) ] ) ( [ ( h { x ) ] ) g [ ( h{ x ) ] h x ( { ) Repeated use of the Chain Rule Example 12 ([ ]) Find f (x) if f (x) = sin cos 4x d dx ([ ]) sin cos 4 x ([ ]) d dx cos cos 4 x cos ( 4x ) cos cos 4 x cos cos 4x ([ ]) sin(4 x) d dx 4x ( ) ( sin(4 x) )( 4 ) 4 cos cos 4x ( ) sin ( 4x ) an alternate notation for example 12 ( ) Find y if y = sin cos 4 x if u = cos 4x y = sin u then where u = cos 4x 6 the der 47 of 4 sin 48 u the 6 der 47 of 4 cos 8 4x cos cos 4x sin 4x ( ) 4 cos cos 4x ( ) sin ( 4 x ) the der of 4x 4 Math 400 3 4B Chain Rule Page 7 of 11 2013 Eitel
Example 13 Repeated use of the Chain Rule Find f (x) if f (x) = ( cos ( 4 x )) 3 d dx ( cos ( 4x ))3 ( ) 2 d dx 3 cos 4x cos ( 4x ) 3 cos 4x ( ) 2 sin(4 x) d dx 4 x 3 cos 4x ( ) 2 sin(4 x) 12 cos 4x 4 ( ) 2 sin 4x an alternate notation for example 13 Find y if y = ( cos ( 4 x )) 3 if u = cos ( 4x ) then y = u 3 where u = cos 4x 6 the 4 der 7 of 4 (u) 48 n ( ) 2 3 cos 4x 12 cos 4 x the 6 der 4 of 7 cos(4 48 x) sin 4x ( ) 2 sin 4 x the der of 4x 4 Math 400 3 4B Chain Rule Page 8 of 11 2013 Eitel
Example 14 The Chain Rule with the Product Rule Find f (x) if f (x) = 4 x 3x 2 2 = 4x ( 3x 2 2) 1/2 644 the first der 7 of 4 sec 448 1 4 x 2 3x2 2 1/2 6x 64 the 7 sec 48 1/2 + 3x 2 2 der of first 4 12x 2 ( 3x 2 2) 1/2 + 4( 3x 2 2) 1/2 factor out a ( 3x 2 2) 1/2 1/2 12x 2 + 4 ( 3x 2 2) 3x 2 2 [ ] [ ] 1/2 12x 2 +12x 2 8 3x 2 2 24 x2 8 3x 2 2 Math 400 3 4B Chain Rule Page 9 of 11 2013 Eitel
Example 15 The Chain Rule with the Quotient Rule Find f (x) if f (x) = 2x 3x 2 2 2x = ( 3x 2 2) 1/2 6 the 4 bottom 748 ( 3x 2 2) 1/2 der of top the top 64 der 4 of 7 bottom 4448 2 2x 1 2 ( 3x2 2) 1/2 6x ( 3x 2 2) 2 14 243 bottom squared 1/2 6x 2 ( 3x 2 2) 1/2 ( 3x 2 2) 2 2 3x 2 2 factor out a ( 3x 2 2) 1/2 [ 6x 2 ] ( 3x 2 2) 2 ( 3x 2 2) 1/2 2 3x 2 2 [ ] 5/2 6x 2 4 6x 2 3x 2 2 4 ( 3x 2 2) 5/2 Math 400 3 4B Chain Rule Page 10 of 11 2013 Eitel
Example 15 The Chain Rule with the Power Rule and the Quotient Rule Find f (x) if f (x) = x 2 2 2 2x + 3 6 the 4 derof 74 u4 n 8 the 64 bottom 74 8 2 x 2 1 2 ( 2x + 3) 2x + 3 der of top 64 the 7 top 48 2x x 2 2 ( 2x + ) 14 243 2 bottom squared der of bottom 2 2 x 2 2 2x + 3 4x 2 + 6x 2x 2 + 4 2x + 3 2 2 x 2 2 2x + 3 2x 2 + 6x + 4 ( 2x + 3) 2 4 x2 +12x + 8 2x + 3 3 Math 400 3 4B Chain Rule Page 11 of 11 2013 Eitel