14 Derivative Problems : Product Rule, Quotient Rule, Chain Rule, First and Second Derivative and Finding Maximum or Minimum

Rules and Applications of the Derivative
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1. Use the Product Rule to find the derivatives of the following functions:

a. f(X) = (1- X^2)*(1+100X)

b. f(X) = (5X + X^-1)*(3X + X^2)

c. f(X) = (X^.5)*(1-X)

d. f(X) = (X^3 + X^4)*(30 + X^2)

2. Use the Chain Rule to find the derivatives of the following functions:

a. f(X) = (1- X^2)^5

b. f(X) = (5X + X^-1)^-1

c. f(X) =(1-X)^2

d. f(X) = (X^3 + X^4)^3

3. Use the Quotient Rule to find the derivatives of the following functions:

a. f(X) = 100/X^4

b. f(X) = 1/(5X + X^2)

c. f(X) =5/(1-X)

4. For each of the following functions find the 1) first and second derivative, 2) explain whether or not the function has a maximum or a minimum, and how you reached that conclusion, and 3) the value of the maximum or minimum

a. f(X) = 5X^2 - 2X

b. f(X) = 1000X - X^2

c. f(X) = 8X^3 - 4X^2

sorry about that forgot the powers

Solution Summary

Fourteen derivative problems are solved regarding Product Rule, Quotient Rule, Chain Rule, First and Second Derivative and Finding Maximum or Minimum. The solution is detailed and well presented.

In each of the following parts, you should simplify your answers where it is appropriate to do so.
(1) (a) Write down the derivative of each of the functions
f(x) = e^-8x and g(x)=sin(4x)
(b) Hence, by using the ProductRule, diļ¬erentiate the function
k(x) = e^-8xsin (4x)
(2) (a) Write down the derivative of eac

1.
a. use quotient rule to find derivative of this function. f(x) = (20+16x-x^2)/(4+x^2).
b. Find any stationary points of the function from 1a. An use the firstderivative test to see whether they are local maximum or local minimum of f(x).
c. what are the maximumandminimum values of the function f(x) at interval [-6,2]

1. For the following function find the value of the derivative at the specific point given using:
- the definition of the derivative
- the sum rule for derivatives
Show that both methods lead to the same result.
f(x)=-x3+3x2-2 at x=1
2. Find an equation to the tangent at the given point, using the Product Rule

Find the point(s) on the graph where the tangent line is horizontal...
50. Extend the product rule for differentiation to the following case involving the product of three differentiable functions...
Find derivative...
(Please see attached.)

Please show all the steps to calculate the answers to the attached questions about using the chain rule on an American population function and a demand function for desk lamp prices.

Need help finding partial derivatives of attached problems.
Find three partial derivatives of the function r with respect to x, y, and z
1. r = uvw − u^2 − v^2 − w^2 where u = y + z, v = x + z,w = x + y
2. r = p / q + q / s + s / p where p = e^yz, q = e^xz, s = e^xy

Let f: R^3 --> R^2 by given by
f((x,y,z)) = (x^2 + y^2*z^2 + y*z^2)
and let g: R^2 --> R^2 be given by
g((u,v)) = (log(1 + u^2v^2), u^3v)
Use the Chain Rule to compute (g o f)' at (1,0,1)