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# the chain rule

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A 16-ft ladder leans against a wall. The bottom of the ladder is 5 ft from
the wall at time t = 0 and slides away from the wall at a rate of 3 ft/s.
Find the velocity of the top of the ladder at time t = 1.
a) The textbook response to this question is - sqrt (3) is approximately equal too
-1.732 ft/s. The minus sign
means the top of the ladder is sliding down. Check that the textbook's answer is
correct.
b) The speed of sound at sea level using the standard atmosphere is about 340.29
meters per second. There are 3.280840 feet in one meter. Using the assumptions
of this model, fi nd the angle between the ladder and the ground at the time that
the top of the ladder breaks the speed of sound.
c) The speed of light is about 299,792,458 meters per second. There are still
3.280840 feet in one meter. Using the assumptions of this model, fi nd the an-
gle between the ladder and the ground at the time that the top of the ladder moves
at the speed of light.

https://brainmass.com/math/calculus-and-analysis/the-chain-rule-387356

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Pythagoras theorem is also reiterated in order to solve.

\$2.19

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

Rules and Applications of the Derivative
--------------------------------------------------------------------------------
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

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