Answer:
-2 ≤ n < 8
Step-by-step explanation:
_________________
Answer:
The purchase price is $180
The total price is: $189
Step-by-step explanation:
Sales tax rate = 5\%
Sales tax on bike = $9
We need to find the purchase price.
Let purchase price = x
We know that Sales tax = 5% of x
and Total price = Purchase Price + Sales tax
So, we can write:
Sales tax = 5% of x
Putting values and finding x
So, the value of x i.e Purchase price = $180
The purchase price is $180
Now finding the total price of bicycle
Total price = Purchase Price + Sales tax
Total price = 180 + 9
Total price = $189
So, The total price is: $189
This is quite a complex problem. I wrote out a really nice solution but I can't work out how to put it on the website as the app is very poorly made. Still, I'll just have to type it all in...
Okay so you need to use a technique called logarithmic differentiation. It seems quite unnatural to start with but the result is very impressive.
Let y = (x+8)^(3x)
Take the natural log of both sides:
ln(y) = ln((x+8)^(3x))
By laws of logarithms, this can be rearranged:
ln(y) = 3xln(x+8)
Next, differentiate both sides. By implicit differentiation:
d/dx(ln(y)) = 1/y dy/dx
The right hand side is harder to differentiate. Using the substitution u = 3x and v = ln(x+8):
d/dx(3xln(x+8)) = d/dx(uv)
du/dx = 3
Finding dv/dx is harder, and involves the chain rule. Let a = x+ 8:
v = ln(a)
da/dx = 1
dv/da = 1/a
By chain rule:
dv/dx = dv/da * da/dx = 1/a = 1/(x+8)
Finally, use the product rule:
d/dx(uv) = u * dv/dx + v * du/dx = 3x/(x+8) + 3ln(x+8)
This overall produces the equation:
1/y * dy/dx = 3x/(x+8) + 3ln(x+8)
We want to solve for dy/dx, achievable by multiplying both sides by y:
dy/dx = y(3x/(x+8) + 3ln(x+8))
Since we know y = (x+8)^(3x):
dy/dx = ((x+8)^(3x))(3x/(x+8) + 3ln(x+8))
Neatening this up a bit, we factorise out 3/(x+8):
dy/dx = (3(x+8)^(3x-1))(x + (x+8)ln(x+8))
Well wasn't that a marathon? It's a nightmare typing that in, I hope you can follow all the steps.
I hope this helped you :)
<span><span>SPOTLIGHT DEMYSTIFIED QUIZZES GALLERIES LISTS ON THIS DAY BIOGRAPHIES</span>SEARCH BRITANNICAWhat are you looking for?SearchBROWSE POPULAR TOPICS:<span>Pearl Harbor attack American Revolution Reign of Terror Photosynthesis Las Posadas</span></span><span>You have reached Britannica's public website.<span>For ad-free access to your Britannica School or Library account CLICK HERE</span></span><span>Chemical compound<span>WRITTEN BY: </span><span>Melvyn C. UsselmanRichard O.C. NormanSteven S. ZumdahlCarl R. Noller</span>See Article History<span><span>Chemical compound, any substance composed of identical moleculesconsisting of atoms of two or more chemical elements.<span><span>Methane, in which four hydrogen atoms are bound to a single carbon atom, is an example of a basic chemical compound. The structures of chemical compounds are influenced by complex factors, such as bond angles and bond length.Encyclopædia Britannica, Inc.</span></span>All the matter in the universe is composed of the atoms of more than 100 different chemical elements, which are found both in pure form and combined in chemical compounds. A sample of any given pure element is composed only of the atoms characteristic of that element, and the atoms of each element are unique. For example, the atoms that constitute carbon are different from those that make up iron, which are in turn different from those of gold. Every element is designated by a unique symbol consisting of one, two, or three letters arising from either the current element name or its original (often Latin) name. For example, the symbols for carbon, hydrogen, and oxygen are simply C, H, and O, respectively. The symbol for iron is Fe, from its original Latin name ferrum. The fundamental principle of the science of chemistry is that the atoms of different elements can combine with one another to form chemical compounds. Methane, for example, which is formed from the elements carbon and hydrogen in the ratio four hydrogen atoms for each carbon atom, is known to contain distinct CH4 molecules. The formula of a compound—such as CH4—indicates the types of atoms present, with subscripts representing the relative numbers of atoms (although the numeral 1 is never written).<span /></span></span><span><span>
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