[Notes & HW Answers] 3.3 The Product and Quotient Rules

 

[Notes & HW Answers] 3.3 The Product and Quotient Rules.pdf

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[Prepwork 3.3]

Q1. Which of the following are possible formulas for a function

y = f (x) such that f'(x) = 10x^9*e^x+x^10*e^x?

• A. x10ex+10

• B. 3+x10ex

• C. x10ex+1

• D. 9x9ex−1

• E. x10ex+10

• F. 10x10ex

• G. x10ex

A: ABG

Q2. Which (single) rule would one use to differentiate f (x) =x*e^x?

A: the Product rule

[HW 3.3]

Q1. Find the derivative of the function f (x), below. It may be to your advantage to simplify first.

f(x) = (x^5−x^(1/2))*9^x

A: f'(x) = (5x^4+x^5(ln 9)- (1/2)x^(-1/2)-x^(1/2)(ln 9)*9^x

Q2. Find the derivative of the function y, below. It may be to your advantage to simplify first.

y =(t +10)/2^t

A: dy/dt = (2^t - (t+10)(ln 2)(2^t))/2^(2t) 

 

Q3. Find the derivative of the function f(z), below. It may be to your advantage to simplify first.

f(z) = (z^2+10) / z^(1/2)

A: f'(z) = (2z*z^(1/2) - (z^2+10)(1/2)z^(-1/2))/z

 

Q4. Find the derivative of the function w(x), below. It may be to your advantage to simplify first.

w(x) = 23e^x/3^x

A: w'(x) = 23e^x*3^(-x)*(1-ln 3)

 

Q5. Use the figure below to estimate the indicated derivatives, or state that they do not exist. If a derivative does not exist, enter dne in the answer blank. The graph of f (x) is black and has a sharp corner at x = 2. The graph of g(x) is blue.

Let h(x) = f(x) ·g(x). Find

 

A:

A. h'(1) = 1

B. h'(2) = dne

C. h'(3) = -1 

 

Q6. Let h(x) = f (x) ·g(x), and k(x) = f (x)/g(x). Use the figures below to find the exact values of the indicated derivatives.

A:

A. h'(2) = -8/3

B. k'(−2) = 2

 

Q7. Let F(3) = 4,F'(3) = 5,H(3) = 4,H'(3) = 2.

A:

A. If G(z) = F(z) ·H(z), then G'(3) = 28

B. If G(w)=F(w)/H(w), then G'(3)= 3/4