Answer:
![\displaystyle f'(x) = \bigg( \frac{1}{2\sqrt{x}} - \sqrt{x} \bigg)e^\big{-x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7D%20%5Cbigg%29e%5E%5Cbig%7B-x%7D)
General Formulas and Concepts:
<u>Algebra I</u>
Terms/Coefficients
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]: ![\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
![\displaystyle f(x) = \frac{\sqrt{x}}{e^x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B%5Csqrt%7Bx%7D%7D%7Be%5Ex%7D)
<u>Step 2: Differentiate</u>
- Derivative Rule [Quotient Rule]:
![\displaystyle f'(x) = \frac{(\sqrt{x})'e^x - \sqrt{x}(e^x)'}{(e^x)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7B%28%5Csqrt%7Bx%7D%29%27e%5Ex%20-%20%5Csqrt%7Bx%7D%28e%5Ex%29%27%7D%7B%28e%5Ex%29%5E2%7D)
- Basic Power Rule:
![\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}(e^x)'}{(e^x)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7B%5Cfrac%7Be%5Ex%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7D%28e%5Ex%29%27%7D%7B%28e%5Ex%29%5E2%7D)
- Exponential Differentiation:
![\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x}{(e^x)^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7B%5Cfrac%7Be%5Ex%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7De%5Ex%7D%7B%28e%5Ex%29%5E2%7D)
- Simplify:
![\displaystyle f'(x) = \frac{\frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x}{e^{2x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7B%5Cfrac%7Be%5Ex%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7De%5Ex%7D%7Be%5E%7B2x%7D%7D)
- Rewrite:
![\displaystyle f'(x) = \bigg( \frac{e^x}{2\sqrt{x}} - \sqrt{x}e^x \bigg) e^{-2x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cbigg%28%20%5Cfrac%7Be%5Ex%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7De%5Ex%20%5Cbigg%29%20e%5E%7B-2x%7D)
- Factor:
![\displaystyle f'(x) = \bigg( \frac{1}{2\sqrt{x}} - \sqrt{x} \bigg)e^\big{-x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cbigg%28%20%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20-%20%5Csqrt%7Bx%7D%20%5Cbigg%29e%5E%5Cbig%7B-x%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Write down
15-80
Solve it by adding 15 to -80
The final answer is -65
Answer:
(0,-2)
Step-by-step explanation:
the answer is (0,-2) because -2 is where the line crosses over the y-axis. If we looked at a graph you would see that the line goes over 0 on the X-axis, which causes the line to touch the y-axis and intercept it at the -2 mark.
An easy way to find the answer for these sorts of questions is to draw out a graph and mark the coordinates on it and draw the line conecting them. Once you've done that you can easily see where the line intercepts on both the y and x axis.
.2 is 2/10 while .5 is 1/2. 1/2 is much larger than 1/5 (which is 2/10 reduced).