Topic 8 The Chain Rule By: Kelley Borgard Block 4A

Theorem 2.10 If y = f(u) is a differentiable function of u, and u = g(x) is a differentiable function of x, then y= f(g(x)) is a differentiable function of x and: If y = f(u) is a differentiable function of u, and u = g(x) is a differentiable function of x, then y= f(g(x)) is a differentiable function of x and: dy/dx= dy/du  du/dx Or d/dx [f(g(x))] =f’(g(x))g’(x)

Example f(x) = 3(x 2 + 2) 3 f(x) = 3(x 2 + 2) 3 Step 1: Substitute an “x” for (x 2 + 2) and take derivative. f(x) = 3(x) 3 f’(x)= 9(x) 2 Step 2: Replace the original values inside the parentheses. f’(x) = 9(x 2 +2) 2 Step 3: Take the derivative of the values inside the parentheses and multiply the result by the first derivative taken. x f’(x) = 2x f’(x) = 9(x 2 +2) 2 2x =18x(x 2 = 2) 2 =18x(x 2 = 2) 2

Try This f(x) = 4(x 2 +1) 3 f(x) = 4(x) 3 (substitute x) f(x) = 4(x) 3 (substitute x) f’(x)= 12(x) 2 f’(x)= 12(x) 2 f’(x) = 12(x 2 +1) 2 (replace with original values) f’(x) = 12(x 2 +1) 2 (replace with original values) f(x) = x 2 +1 → f’(x) = 2x (take derivative of values inside the parentheses) f(x) = x 2 +1 → f’(x) = 2x (take derivative of values inside the parentheses) f’(x) = 12(x 2 +1) 2 (2x) (multiply together) f’(x) = 12(x 2 +1) 2 (2x) (multiply together) f’(x) = 24x (x 2 +1) 2 f’(x) = 24x (x 2 +1) 2

More Examples Example 1 f(x) = √(2x 2 +4) = (2x 2 +4) 1/2 = (2x 2 +4) 1/2 f’(x) = ½ (2x 2 +4) -1/2 (4x) = 2x/(2x 2 +4) 1/2 = 2x/(2x 2 +4) 1/2 Example 2 f(x) = 2/(x+3) 3 = 2(x+3) -3 f’(x) = -6(x+3) -4 (1) = -6/(x+3) 4

Higher Order Derivatives First Derivative dy/dx d/dx f(x) f’(x)y’ Second Derivative d 2 y/dx 2 d 2 y/dx 2 f(x) f”(x)y’’ Third Derivative d 3 y/dx 3 d 3 y/dx 3 f(x) f’’’(x)y’’’ Fourth Derivative d 4 y/dy 4 d 4 y/dx 4 f(x) f (4) (x) y (4) Nth Derivative d n y/dy n d n y/dx n f(x) f (n) (x) y (n)

Example Find the 1 st, 2 nd, 3 rd, and 4 th derivatives. Then Find all the derivatives up through 100. Find the 1 st, 2 nd, 3 rd, and 4 th derivatives. Then Find all the derivatives up through 100. f(x)=4x 3 f’(x)=12x 2 f’’(x)=24xf’’’(x)=24 f (4) (x)=0 f (5) (x)=0 f (5) (x)=f (6) (x)=…=f (100) (x)=0

Theorems 2.6 and : Derivatives of Sine and Cosine Functions 2.6: Derivatives of Sine and Cosine Functions d/dx [sin x] = cos x d/dx [sin x] = cos x d/dx [cos x] = -sin x d/dx [cos x] = -sin x 2.9: Derivatives of Trigonometric Functions 2.9: Derivatives of Trigonometric Functions d/dx [tan x] = sec 2 x d/dx [tan x] = sec 2 x d/dx [sec x] = sec x tan x d/dx [sec x] = sec x tan x d/dx [cot x] = -csc 2 x d/dx [cot x] = -csc 2 x d/dx [csc x] = -csc x cot x d/dx [csc x] = -csc x cot x

Natural Log Ln (1) = 0 Ln (1) = 0 Ln (ab) = Ln(a) + Ln(b) Ln (ab) = Ln(a) + Ln(b) Ln (a n ) = nLna Ln (a n ) = nLna Ln (a/b) = Lna – Lnb Ln (a/b) = Lna – Lnb * a and b are positive and n is rational