The solution of 4x - 8 + 3x = 49 - 73 + 4x - 1 + 2 is equal to ![x = \frac{-5}{3}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-5%7D%7B3%7D)
<u>Solution:</u>
Given, equation is 4x – 8 + 3x = 49 – 73 + 4x – 1 + 2
We have to solve the above given equation for x.
Now, take the given equation.
4x – 8 + 3x = 49 – 73 + 4x – 1 + 2
Adding the like terms we get,
7x – 8 = - 14 + 4x + 1
7x – 4x = - 14 + 1 + 8
3x = - 14 + 9
3x = - 5
![x = \frac{-5}{3}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-5%7D%7B3%7D)
Hence, the value of x is ![\frac{-5}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B-5%7D%7B3%7D)
Answer: See explanation
Step-by-step explanation:
We should note that the value of the digit 2 in the number 32,000 is 2 thousand. The value of the digit 2 in the number 26,000 is 20 thousand = 20,000.
We can simply say that the value of the digit 2 in 26000 is ten times the value of the digit 2 in 32000.
Answer:
![\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Algebra I</u>
- Functions
- Function Notation
- Exponential Rule [Root Rewrite]:
![\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
<u>Algebra II</u>
- Logarithms and Natural Logs
- Logarithmic Property [Multiplying]:
![\displaystyle log(ab) = log(a) + log(b)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20log%28ab%29%20%3D%20log%28a%29%20%2B%20log%28b%29)
- Logarithmic Property [Exponential]:
![\displaystyle log(a^b) = b \cdot log(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20log%28a%5Eb%29%20%3D%20b%20%5Ccdot%20log%28a%29)
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
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)
Logarithmic Derivative: ![\displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Blnu%5D%20%3D%20%5Cfrac%7Bu%27%7D%7Bu%7D)
Implicit Differentiation
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<em />
<em />
<em />
<u>Step 2: Rewrite</u>
- [Equality Property] ln both sides:
![\displaystyle lny = ln(x\sqrt[3]{1 + x^2})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%29)
- Logarithmic Property [Multiplying]:
![\displaystyle lny = ln(x) + ln(\sqrt[3]{1 + x^2})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20ln%28%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%29)
- Exponential Rule [Root Rewrite]:
![\displaystyle lny = ln(x) + ln \bigg[ (1 + x^2)^\bigg{\frac{1}{3}} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20ln%20%5Cbigg%5B%20%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B1%7D%7B3%7D%7D%20%5Cbigg%5D)
- Logarithmic Property [Exponential]:
![\displaystyle lny = ln(x) + \frac{1}{3}ln(1 + x^2)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20lny%20%3D%20ln%28x%29%20%2B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29)
<u>Step 3: Differentiate</u>
- ln Derivative [Implicit Differentiation]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx} \bigg[ ln(x) + \frac{1}{3}ln(1 + x^2) \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20ln%28x%29%20%2B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29%20%5Cbigg%5D)
- Rewrite [Derivative Property - Addition]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{d}{dx} \bigg[ \frac{1}{3}ln(1 + x^2) \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%29%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B3%7Dln%281%20%2B%20x%5E2%29%20%5Cbigg%5D)
- Rewrite [Derivative Property - Multiplied Constant]:
![\displaystyle \frac{d}{dx}[lny] = \frac{d}{dx}[ln(x)] + \frac{1}{3}\frac{d}{dx}[ln(1 + x^2)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blny%5D%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%28x%29%5D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%5Cfrac%7Bd%7D%7Bdx%7D%5Bln%281%20%2B%20x%5E2%29%5D)
- ln Derivative [Chain Rule]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \frac{d}{dx}[(1 + x^2)]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D%5B%281%20%2B%20x%5E2%29%5D)
- Rewrite [Derivative Property - Addition]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot \bigg( \frac{d}{dx}[1] + \frac{d}{dx}[x^2] \bigg)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%5Cbigg%28%20%5Cfrac%7Bd%7D%7Bdx%7D%5B1%5D%20%2B%20%5Cfrac%7Bd%7D%7Bdx%7D%5Bx%5E2%5D%20%5Cbigg%29)
- Basic Power Rule]:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot (2x^{2 - 1})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%20%282x%5E%7B2%20-%201%7D%29)
- Simplify:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{1}{3} \bigg( \frac{1}{1 + x^2} \bigg) \cdot 2x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%29%20%5Ccdot%202x)
- Multiply:
![\displaystyle \frac{y'}{y} = \frac{1}{x} + \frac{2x}{3(1 + x^2)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7By%27%7D%7By%7D%20%3D%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D)
- [Multiplication Property of Equality] Isolate <em>y'</em>:
![\displaystyle y' = y \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20y%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- Substitute in <em>y</em>:
![\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{1}{x} + \frac{2x}{3(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7Bx%7D%20%2B%20%5Cfrac%7B2x%7D%7B3%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- [Brackets] Add:
![\displaystyle y' = x\sqrt[3]{1 + x^2} \bigg[ \frac{5x^2 + 3}{3x(1 + x^2)} \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20x%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%20%5Cbigg%5B%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3x%281%20%2B%20x%5E2%29%7D%20%5Cbigg%5D)
- Multiply:
![\displaystyle y' = \frac{(5x^2 + 3)\sqrt[3]{1 + x^2}}{3(1 + x^2)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B%285x%5E2%20%2B%203%29%5Csqrt%5B3%5D%7B1%20%2B%20x%5E2%7D%7D%7B3%281%20%2B%20x%5E2%29%7D)
- Simplify [Exponential Rule - Root Rewrite]:
![\displaystyle y' = \frac{5x^2 + 3}{3(1 + x^2)^\bigg{\frac{2}{3}}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B5x%5E2%20%2B%203%7D%7B3%281%20%2B%20x%5E2%29%5E%5Cbigg%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Implicit Differentiation
Book: College Calculus 10e
Answer:
55.46
Step-by-step explanation:
I'm guessing you meant 47% of 118
So the answer would be 55.46.
Ok so if u know a percent you multiply but you turn your 75 to a decimal so it would be 0.75•18