Topic 8Topic 8The Chain RuleThe Chain Rule

By: Kelley BorgardBy: Kelley Borgard

Block 4ABlock 4A

Theorem 2.10Theorem 2.10

If y = f(u) is a differentiable function If y = f(u) is a differentiable function of u, and u = g(x) is a differentiable of u, and u = g(x) is a differentiable function of x, then y= f(g(x)) is a function of x, then y= f(g(x)) is a differentiable function of x and:differentiable function of x and:

dy/dx= dy/du dy/dx= dy/du du/dx du/dx

OrOr

d/dx [f(g(x))] =f’(g(x))g’(x)d/dx [f(g(x))] =f’(g(x))g’(x)

ExampleExample f(x) = 3(xf(x) = 3(x22 + 2) + 2)33

Step 1: Substitute an “x” for (xStep 1: Substitute an “x” for (x22 + 2) and take + 2) and take derivative.derivative.

f(x) = 3(f(x) = 3(xx))33

f’(x)= 9(f’(x)= 9(xx))22

Step 2: Replace the original values inside the Step 2: Replace the original values inside the parentheses.parentheses.

f’(x) = 9(f’(x) = 9(xx22 +2+2))22

Step 3: Take the derivative of the values inside the Step 3: Take the derivative of the values inside the parentheses and multiply the result by the first parentheses and multiply the result by the first derivative taken.derivative taken.

xx22 + 2 + 2f’(x) = f’(x) = 2x2x

f’(x) = 9(xf’(x) = 9(x22 +2) +2)2 2 2x2x =18x(x=18x(x22 = 2) = 2)22

Try ThisTry This

f(x) = 4(xf(x) = 4(x22 +1) +1)33

f(x) = 4(f(x) = 4(xx))3 3 (substitute x)(substitute x) f’(x)= 12(f’(x)= 12(xx))22

f’(x) = 12(f’(x) = 12(xx22 +1+1))2 2 (replace with original (replace with original values)values)

f(x) = xf(x) = x22 +1 +1 →→ f’(x) = 2x f’(x) = 2x (take derivative (take derivative of values inside the parentheses)of values inside the parentheses)

f’(x) = 12(xf’(x) = 12(x22 +1) +1)22 (2x) (2x) (multiply together)(multiply together)

f’(x) = 24x (xf’(x) = 24x (x22 +1) +1)2 2

More ExamplesMore Examples

Example 1Example 1

f(x) = √(2xf(x) = √(2x22 +4) +4)

= (2x= (2x22 +4) +4)1/21/2

f’(x) = ½ (2xf’(x) = ½ (2x22+4)+4)-1/2-1/2 (4x)(4x)

= 2x/(2x= 2x/(2x22+4)+4)1/21/2

Example 2Example 2

f(x) = 2/(x+3)f(x) = 2/(x+3)33

= 2(x+3)= 2(x+3)-3-3

f’(x) = -6(x+3)f’(x) = -6(x+3)-4-4 (1) (1)

= -6/(x+3)= -6/(x+3)44

Higher Order DerivativesHigher Order Derivatives

First First DerivatiDerivativeve

dy/dxdy/dx d/dx f(x)d/dx f(x) f’(x)f’(x) y’y’

Second Second DerivatiDerivativeve

dd22y/dxy/dx22 dd22y/dxy/dx22 f(x)f(x)

f”(x)f”(x) y’’y’’

Third Third DerivatiDerivativeve

dd33y/dxy/dx33 dd33y/dxy/dx33 f(x)f(x)

f’’’(x)f’’’(x) y’’’y’’’

Fourth Fourth DerivatiDerivativeve

dd44y/dyy/dy44 dd44y/dxy/dx44 f(x)f(x)

ff(4)(4)(x)(x) yy(4)(4)

Nth Nth DerivatiDerivativeve

ddnny/dyy/dynn ddnny/dxy/dxnn f(x)f(x)

ff(n)(n)(x)(x) yy(n)(n)

ExampleExample Find the 1Find the 1stst, 2, 2ndnd, 3, 3rdrd, and 4, and 4thth derivatives. derivatives.

Then Find all the derivatives up through Then Find all the derivatives up through 100.100.

f(x)=4xf(x)=4x33

f’(x)=12xf’(x)=12x22

f’’(x)=24xf’’(x)=24xf’’’(x)=24f’’’(x)=24f f (4)(4)(x)=0(x)=0ff(5)(5)(x)=0(x)=0ff(5)(5)(x)=f(x)=f(6)(6)(x)=…=f(x)=…=f(100)(100)(x)=0(x)=0

Theorems 2.6 and 2.9Theorems 2.6 and 2.9

2.6: Derivatives of Sine and Cosine 2.6: Derivatives of Sine and Cosine FunctionsFunctions d/dx [sin x] = cos xd/dx [sin x] = cos x d/dx [cos x] = -sin xd/dx [cos x] = -sin x

2.9: Derivatives of Trigonometric 2.9: Derivatives of Trigonometric Functions Functions d/dx [tan x] = secd/dx [tan x] = sec22xx d/dx [sec x] = sec x tan xd/dx [sec x] = sec x tan x d/dx [cot x] = -cscd/dx [cot x] = -csc22xx d/dx [csc x] = -csc x cot xd/dx [csc x] = -csc x cot x

Natural LogNatural Log

Ln (1) = 0Ln (1) = 0 Ln (ab) = Ln(a) + Ln(b)Ln (ab) = Ln(a) + Ln(b) Ln (aLn (ann) = nLna) = nLna Ln (a/b) = Lna – LnbLn (a/b) = Lna – Lnb

** a a and and bb are positive and are positive and nn is rational is rational