Use the chain rule to evaluate the derivative. 1. Find f'(x) if f(x) = (2x - 7)6. 2. Find [(4x3 - 7x) 4 ]. 3. Find [ ( 4X - 7) 4 (7 x + 8 ) 5 ] dx 4. Find d (4x - 7)3 dx (7x +8)3 Evaluate the first, second, and third derivatives of the following functions. 5. Findf(x) = 12x6 - 3x4+2x2. 6. Findf(x) =(2x - 7)6. This one is challenging as you will need to rely on basic derivative rules, the chain rule and the product rule.2. Examples of finding the derivative using the chain rule. a. Here is the rule. (fog)'(x)= f'(g(x)) . g'(x). Find f'(x) b. Written example. f (x) = (x3 -3)2. Find f'(x) f' ( x) = - (x3 - 3)2 = 2(x3 - 3)1_d (x 3 - 3) =2(x3 - 3)(3x2) dx = (2x3 -6)(3x2) 24-7 =6x6 - 18x2 om2. Examples of finding the derivative using the chain rule. a. Here are the rules we will be using. CHAIN RULE: (f . g)'(x) = f'(g(x)) - g'(x). Find f'(x) PRODUCT RULE: (f(x) . g(x))' =f(x)g'(x)+f'(x)g(x) b. Written example. f(x) = (5x+2)2(4x -3)3. Find f'(x) 1. Let's apply the product rule first. f' ( x) = - [(5x + 2)2(4x - 3)3] = -[(5x+2)2](4x- 3)3 + (5x + 2)2-[(4x-3)3] dx dx dx -7 2. Now we need to use the chain rule to find the derivates within step 1. = 2(5x + 2)(5)(4x - 3)3 + (5x +2)2[3(4x-3)2(4)] 3. Simplify. = (5x + 2)(4x - 3)2(2(5)(4x - 3)1+ (5x+ 2)*(3)(4)) THIS IS COMPLICATED AS I AM FACTORING OUT COMMON BINOMIAL FACTORS. = (5x + 2)(4x - 3)2(10(4x - 3)+ 12(5x +2)) = (5x + 2)(4x - 3)2((40x - 30) + (60x+24)) = (5x +2)(4x - 3)2(100x -6)2. Examples of finding the derivative using the chain rule. a. Here are the rules we will be using. CHAIN RULE: (f . g)'(x) = f'(g(x)) . g'(x). Find f'(x) QUOTIENT RULE: d ( f(x) \\_f'(x)g(x) -f(x)g' (x) OR- ax [f(x)]g(x) -f(x)-[g(x)] dx ( g(x) (g(x))2 (g(x))2 b. Written example. f(x) = (5x+ 2)2 (4x - 3)3 . Find f'(x) 1. Apply the quotient rule. f'(x) = d (5x+2)2 1 ax [(5x + 2)2](4x -3)3 -(5x+2)2-d [(4x -3)3] dx (4x - 3)3 ((4x - 3) 3)2 2. Apply the chain rule. [(5x + 2)21(4x -3)3-(5x+2)2 [(4x-3)3] [2(5x+ 2)(5)1(4x - 3)3 -(5x +2)2[3(4x-3)2(4)] ((4x - 3)3)2 ((4x -3) 3)2 3. Simplify using factoring. [10(5x + 2)](4x - 3)3 - (5x + 2)2[12(4x -3)2] _ (5x + 2)(4x -3)2[10(4x - 3)1 - (5x +2)1(12)] (4x - 3) 6 (4x - 3)6 (5x + 2)(4x - 3)2[(40x-30)- (60x + 24)] (4x - 3)6 (5x + 2)(4x -3)2[-40x-54] (4x - 3)6 Cancel like factors in numerator and denominator: (5x + 2)(4x-3)2[- 40x - 54] _ (5x + 2)(4x-3)2[-40x-54] (4x - 3) 6 (4x 3)2 (4x - 3)4 4. State the derivative. f'(x)= (5x+2)(-40x -54) (4x - 3)42. Examples of finding the derivative using the chain rule. a. Here are the rules we will be using. 1st derivative: f'(x) 2nd derivative: f"(x) 3rd derivative: f"(x) OR f(3)(x) We can continue taking derivatives as long as they exist, but we will stop at the third derivative for now. b. Written example. f(x)= - 2x5 +3x2 +5. Find the first, second, and third derivatives. 1. First derivative. f'(x)= - 10x4+6x 2. Second derivative. This is the derivative of the first derivative. f"(x) = [-10x4+ 6x]= -40x3 +6 3. Third derivative. This is the derivative of the second derivative. f""(x)= -[-40x3 +6]= -120x2 dxQuestion 1 7 pts Use the chain rule to evaluate. ( (2 x - 1) 6 ) dx For your answer, use the following as an example: Type 6(x-5)^3 for 6(x -5)3 Question 2 8 pts Use the product rule along with the chain rule to evaluate. d ( x 3 ( 2 x - 3) 2 ) dx For your answer, use the following as an example: Type 6x^2(x-5)^3(x-10) for 6x2(x - 5)3(x -10)