1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alex
3 years ago
11

Find dy/dx if f(x) = (x + 8)^3x.

Mathematics
1 answer:
vlabodo [156]3 years ago
8 0
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 :)
You might be interested in
What is the sum of all exterior angles of a regular octagon
8090 [49]
The 8 exterior angles of a regular octagon add up to 360 degrees
3 0
2 years ago
Identify the terms, coefficients, and constants in the expression. 7h + 3
dexar [7]

Answer:

7h+10

Step-by-step explanation:


7 0
3 years ago
Skip and Jena are selling raffle tickets to raise money for new basketball uniforms. Skip sold 2 tickets for every 7 tickets tha
user100 [1]

Answer:

gidetgfddtgfft5f5f5f5d5f6guhuhug5xtdrd4s4d5d6v7g7h8n8b7v6v6

7 0
3 years ago
Which subsets of real numbers does -22 belong
deff fn [24]
Negative number because -22
3 0
3 years ago
Please answer tomorrow is my exam please​
Illusion [34]

Answer:

i dont know sorry.

Step-by-step explanation:

Do it yourself ;)

4 0
3 years ago
Other questions:
  • Draw a graph of 2x - 3y = 6​
    10·1 answer
  • Ms. Toyama bought a total of 2/3 ton of white and brown landscape rocks. The weight of the white and brown landscape rocks. The
    14·1 answer
  • an adult takes 600mg of ibuprofen. Each hour, the amount of ibuprofen in the persons system decreases by about 29%. About how mu
    8·1 answer
  • What is the exponential notation of 100
    10·1 answer
  • What is the common denominator for 14/25 and 10/21
    6·1 answer
  • URGENT PLEASE ANSWER To find the distance across a river, a surveyor chooses points A and B on one side of the river. She then c
    7·1 answer
  • WILL MARK BRAINLIEST EASY MATH PROBLEM!!! + FREE VBUCKSSS
    15·2 answers
  • for what value of the variable is the value of the expression -3(2x+1) is 20 greater than the expression 8x+5
    9·1 answer
  • Jeremie bought 5 pounds of bananas for $4.45. How much is 1 pound of bananas? Show or explain your reasoning
    9·2 answers
  • What is the second term of the sequence generated by the fo02O 3O 5O 6
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!