Work out 4 x 15 7. Give your answer as a mixed number.

vlabodo [156]

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 :)

8 0

3 years ago

Convert 32 ½ meters to feet

Olin [163]

Answer: 32 1/2 meters = 106.63 ft

Step-by-step explanation: 1 meter is 3 ft and 3.37 inches, or 39.37 inches :)

5 0

2 years ago

Read 2 more answers

Please help me!! (x+5) (x-2) Any work out or help would be appreciated!! Thanks! :)

Lostsunrise [7]

Answer:

Step-by-step explanation:

If you treat (x+5) as a single value and (x-2) as the values of x and +2, you can distribute (x+5) to x and +2: ( (x-5)*x + (x-5)*2 ). Then you can treat x-5 as the values of x and -5 and distribute those to the numbers next to them: (×^2 - 5x) + (2x - 10). This results in x^2 -3x - 10.

4 0

3 years ago

Yesterday Mike bought 2 gallons of regular gasoline and 3 gallons of premium gasoline at a gas station for $13.60 today he bough

larisa [96]

Answer:

Cost of 1 gallon of Regular gasoline is $2.45 and Cost of 1 gallon of Premium gasoline is $2.90.

Step-by-step explanation:

Let the Cost of Regular gasoline be 'x'.

Let the Cost of Premium gasoline be 'y'.

Given:

Amount of regular gasoline bought yesterday = 2 gallons

Amount of premium gasoline bought yesterday = 3 gallons

Total Cost of yesterday = $13.60

Now we know that Total Cost of yesterday is equal to sum of Amount of regular gasoline bought yesterday multiplied by Cost of Regular gasoline and Amount of Premium gasoline bought yesterday multiplied by Cost of premium gasoline.

Framing in equation form we get;

$2x+3y=13.60 \ \ \ \ equation\ 1$

Also Given:

Amount of regular gasoline bought today = 3 gallons

Amount of premium gasoline bought Today = 4 gallons

Total Cost of Today = $18.95

Now we know that Total Cost of Today is equal to sum of Amount of regular gasoline bought Today multiplied by Cost of Regular gasoline and Amount of Premium gasoline bought Today multiplied by Cost of premium gasoline.

Framing in equation form we get;

$3x+4y=18.95 \ \ \ \ equation\ 2$