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3 years ago

Simplify the expression, 2+10x+8 *2-16

Mathematics

1 answer:

jasenka [17]3 years ago

7 0

Answer:

2(1+5x)

Step-by-step explanation:

2+10x+8 *2-16

2+10x+0

2+10x

2(1+5x)

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(d). Use an appropriate technique to find the derivative of the following functions:

natima [27]

(i) I would first suggest writing this function as a product of the functions,

$\displaystyle y = fgh = (4+3x^2)^{1/2} (x^2+1)^{-1/3} \pi^x$

then apply the product rule. Hopefully it's clear which function each of f, g, and h refer to.

We then have, using the power and chain rules,

$\displaystyle \frac{df}{dx} = \frac12 (4+3x^2)^{-1/2} \cdot 6x = \frac{3x}{(4+3x^2)^{1/2}}$

$\displaystyle \frac{dg}{dx} = -\frac13 (x^2+1)^{-4/3} \cdot 2x = -\frac{2x}{3(x^2+1)^{4/3}}$

For the third function, we first rewrite in terms of the logarithmic and the exponential functions,

$h = \pi^x = e^{\ln(\pi^x)} = e^{\ln(\pi)x}$

Then by the chain rule,

$\displaystyle \frac{dh}{dx} = e^{\ln(\pi)x} \cdot \ln(\pi) = \ln(\pi) \pi^x$

By the product rule, we have

$\displaystyle \frac{dy}{dx} = \frac{df}{dx}gh + f\frac{dg}{dx}h + fg\frac{dh}{dx}$

$\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} (x^2+1)^{-1/3} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} (x^2+1)^{-1/3} \ln(\pi) \pi^x$

$\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} \frac{1}{(x^2+1)^{1/3}} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} \frac{1}{ (x^2+1)^{1/3}} \ln(\pi) \pi^x$

$\displaystyle \frac{dy}{dx} = \boxed{\frac{\pi^x}{(4+3x^2)^{1/2} (x^2+1)^{1/3}} \left( 3x - \frac{2x(4+3x^2)}{3(x^2+1)} + (4+3x^2)\ln(\pi)\right)}$

You could simplify this further if you like.

In Mathematica, you can confirm this by running

D[(4+3x^2)^(1/2) (x^2+1)^(-1/3) Pi^x, x]

The immediate result likely won't match up with what we found earlier, so you could try getting a result that more closely resembles it by following up with Simplify or FullSimplify, as in

FullSimplify[%]

(% refers to the last output)

If it still doesn't match, you can try running

Reduce[<our result> == %, {}]

and if our answer is indeed correct, this will return True. (I don't have access to M at the moment, so I can't check for myself.)

(ii) Given

$\displaystyle \frac{xy^3}{1+\sec(y)} = e^{xy}$

differentiating both sides with respect to x by the quotient and chain rules, taking y = y(x), gives

$\displaystyle \frac{(1+\sec(y))\left(y^3+3xy^2 \frac{dy}{dx}\right) - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = e^{xy} \left(y + x\frac{dy}{dx}\right)$

$\displaystyle \frac{y^3(1+\sec(y)) + 3xy^2(1+\sec(y)) \frac{dy}{dx} - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = ye^{xy} + xe^{xy}\frac{dy}{dx}$

$\displaystyle \frac{y^3}{1+\sec(y)} + \frac{3xy^2}{1+\sec(y)} \frac{dy}{dx} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} \frac{dy}{dx} = ye^{xy} + xe^{xy}\frac{dy}{dx}$

$\displaystyle \left(\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}\right) \frac{dy}{dx}= ye^{xy} - \frac{y^3}{1+\sec(y)}$

$\displaystyle \frac{dy}{dx}= \frac{ye^{xy} - \frac{y^3}{1+\sec(y)}}{\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}}$

which could be simplified further if you wish.

In M, off the top of my head I would suggest verifying this solution by

Solve[D[x*y[x]^3/(1 + Sec[y[x]]) == E^(x*y[x]), x], y'[x]]

but I'm not entirely sure that will work. If you're using version 12 or older (you can check by running $Version), you can use a ResourceFunction,

ResourceFunction["ImplicitD"][<our equation>, x]

but I'm not completely confident that I have the right syntax, so you might want to consult the documentation.

3 0

2 years ago

Which expression is equivalent to -1/12x-1/3?

hichkok12 [17]

B. If you multiply 1/12 by (-x) it's -1/12x and if you multiply 1/12 by (-4) it's -4/12 which is equivalent to -1/3.

4 0

2 years ago

Read 2 more answers

A triangle has a base of 7 1/8 feet and height 7 1/5 ft. what is the area as a mixed number

sleet_krkn [62]

The answer is 25 13/20 square feet.

In order to complete this operation, it is first best to change your mixed numbers into improper fractions.

7 1/8 = 57/8

7 1/5 = 36/5

Now that we have these, we can put them into the triangle formula.

A = 1/2bh

A = (1/2)(57/8)(36/5)

A = 1026/40

You can then simplify this fraction by dividing the top and bottom by a factor of 2.

A = 513/20

Then, because we know 20 goes into 500, 25 times, we can pull that out and be left with a remainder of 13/20.

A = 25 13/20

8 0

3 years ago

You are school supply shopping in together as a package of six pencils for $2.70 how much does each pencil cost

Darya [45]

Answer:

0.45 cents

Step-by-step explanation:

4 0

3 years ago

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chelsea deposits $600 into a bank account and is earning simple interest.After 4 years her account has a balance of $621.60.If C

olga nikolaevna [1]

If she deposited $600 and had $621.60 after four years, then she earned: $621.60 - $600 = $21.60 in four years. It means in one year she earned $21.60 ÷ 4 = $5.40. Now we have to find out how many percent of $600 is $5.40:

$5.40 = x% from $600
5.4 = x/100 * 600
5.4 = 600x/100 / * 100 (both sides)
540 = 600x / ÷ 600 (both sides)
x = 0.9

So the interest rate was 0.9% annually.

Now we have to find out how much is 0.9% from $1100:

$1100 * 0.9% =
= 1100 * 9/1009 =
= 9900/1000 =
= 9.9

It means in one year she would earn $9.90. After 7 years she would earn $9.90 * 7 = $69.30, so she would have in her account:

$1100 + $69.30 = <u>$1169.30</u>

8 0

3 years ago