Step-by-step explanation:
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Answer:

General Formulas and Concepts:
<u>Algebra I</u>
- Functions
- Function Notation
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Rule [Chain Rule]: ![\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
Derivative: ![\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Be%5Eu%5D%3De%5Eu%20%5Ccdot%20u%27)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em />
<em />
<em />
<u>Step 2: Differentiate</u>
- eˣ Derivative [Derivative Rule - Chain Rule]:
![\displaystyle J'(x) = \frac{d}{dx}[e^{f(x)}] \cdot \frac{d}{dx}[f(x)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20J%27%28x%29%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Be%5E%7Bf%28x%29%7D%5D%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28x%29%5D)
- Simplify:

<u>Step 3: Evaluate</u>
- Substitute in <em>x</em> [Derivative]:

- Substitute in function values:

- Simplify:

Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
If <em>y</em> = cos(√<em>x</em>), then by the chain rule
d<em>y</em>/d<em>x</em> = d/d<em>x</em> [cos(√<em>x</em>)]
d<em>y</em>/d<em>x</em> = - sin(√<em>x</em>) d/d<em>x</em> [√<em>x</em>]
d<em>y</em>/d<em>x</em> = - sin(√<em>x</em>) (1 / (2√<em>x</em>))
d<em>y</em>/d<em>x</em> = - 1/2 sin(√<em>x</em>) / √<em>x</em>
Answer:
382 children and
312 adults swam
Step-by-step explanation:
Let the number of children be x
And let the number of adult be y
Hence on a hot summer day
x +y = 694 - - - - - - - 1
x= 694-y
Also the daily prices are $ 1.25 for children and $ 2.50 for adults
1.25x + 2.50y= 1257.50-------2
Let us substitute x= 694-y in equation 2
1.25(694-y) + 2.50y =1257.50
867.5-1.25y +2.50y = 1257.50
-1.25y+2.50y= 1257.58-867.5
1.25y= 390
y= 398/1.25
y= 312
Substitute y= 312 in equation 1 to find x
x + 312= 694
x= 694-312
x= 382
Hence 382 children and
312 adults swam
Answer: See explanation
Step-by-step explanation:
1. Let the small number be x
Then, the bug number will be: x + 25
Therefore, x + x + 25 = 131
2x + 25 = 131
2x = 131 - 25
2x = 106
x = 106 / 2
x = 53
Small number= 53
Big number = x + 25 = 53 + 25 = 78
2. Let the small number be x
Then, the bug number will be: x + 14
x + x + 14 = 58
2x + 14 = 58
2x = 58 - 14
2x = 44
x = 44/2
x = 22
Small number = 22
Bug number = 22 + 14 = 36
The numbers are 22 and 36.
3. Let number of basketball sold be x
Therefore, number of football sold will be x+3.
Therefore, x + x + 3 = 27
2x + 3 = 27
2x = 27 - 3
2x = 24
x = 24/2
x = 12
12 basketball were sold
Number of footballs sold = 12 + 3 = 15
4. Let calories in bowl of soup x
Hamburger = 2 × x = 2x
Therefore, x + 2x = 738
3x = 738
x = 738/3
x = 246
There are 246 calories in bowl of soup
Calories in hamburgers = 246 × 2 = 492 calories