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
MrRa [10]
3 years ago
12

Find y' if y= In (x2 +6)^3/2 y'=

Mathematics
1 answer:
Schach [20]3 years ago
3 0

Answer:

\displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right<u> </u>

<u>Algebra I</u>

  • Factoring

<u>Calculus</u>

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Derivative Property [Multiplied Constant]:                                                                \displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)

Derivative Rule [Chain Rule]:                                                                                     \displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

ln Derivative: \displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}

Step-by-step explanation:

<u>Step 1: Define</u>

<u />\displaystyle y = ln(x^2 + 6)^{\frac{3}{2}}<u />

<u />

<u>Step 2: Differentiate</u>

  1. [Derivative] Chain Rule:                                                                                 \displaystyle y' = \frac{d}{dx}[ln(x^2 + 6)^{\frac{3}{2}}] \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  2. [Derivative] Chain Rule [Basic Power Rule]:                                                 \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{3}{2} - 1} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  3. [Derivative] Simplify:                                                                                      \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  4. [Derivative] ln Derivative:                                                                               \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot \frac{d}{dx}[x^2 + 6]
  5. [Derivative] Basic Power Rule:                                                                      \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2 \cdot x^{2 - 1} + 0)
  6. [Derivative] Simplify:                                                                                       \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2x)
  7. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2} \cdot \frac{1}{x^2 + 6} \cdot (2x)
  8. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)} \cdot (2x)
  9. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3(2x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  10. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{6xln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  11. [Derivative] Factor:                                                                                         \displaystyle y' = \frac{2(3x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  12. [Derivative] Simplify:                                                                                       \displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e

You might be interested in
Match each quadratic equation with its solution set.
Luba_88 [7]

Answer:

2x²-32 ⇒ x²=16⇒ (-4,4)

4x²-100 ⇒x²=25 ⇒(-5,5)

x²-55=9 ⇒x²=64 ⇒(-8,8)

x²-140=-19 ⇒x²=121 ⇒(-11,11)

2x²-18=0 ⇒x²=9 ⇒(-3,3)

5 0
3 years ago
Read 2 more answers
What are 5 fractions that can simplify to 2/5
enot [183]
4/10
8/20
16/40
32/80
64/160
6 0
3 years ago
Read 2 more answers
Find the function value, if possible. (If an answer is undefined, enter UNDEFINED.)
ddd [48]

Answer:

f(x + 2) = 3x + 2

Step-by-step explanation:

Simply replace wherever you find x in the function f(x) with (x + 2), like so:

f(x + 2) = 3(x + 2) - 4

f(x + 2) = 3x + 6 - 4

f(x + 2) = 3x + 2

7 0
3 years ago
What are the coordinates of the point on the directed line segment from (-6, -3) to
melamori03 [73]

Given:

A point divides a directed line segment from (-6, -3) to (5,8) into a ratio of 6 to 5.

To find:

The coordinates of that point.

Solution:

Section formula: If point divides a line segment in m:n, then the coordinates of that point are

Point=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)

A point divides a directed line segment from (-6, -3) to (5,8) into a ratio of 6 to 5. Using section formula, we get

Point=\left(\dfrac{6(5)+5(-6)}{6+5},\dfrac{6(8)+5(-3)}{6+5}\right)

Point=\left(\dfrac{30-30}{11},\dfrac{48-15}{11}\right)

Point=\left(\dfrac{0}{11},\dfrac{33}{11}\right)

Point=\left(0,3\right)

Therefore, the coordinates of the required point are (0,3).

3 0
3 years ago
PLEASE HELP!! WILL MARK BRAINLIEST IF RIGHT!!!
denpristay [2]

Answer:

images are:

W'(-5,0)

X'(0,-9)

Y'(-9,-6)

Z'(-6,-2)

Step-by-step explanation:

use formula p(x,y)=p'(y,-x)

4 0
2 years ago
Other questions:
  • A football coach recorded his team's game scores over a football seasons. The scores are 21,45,21,14,21,28,24,14,24,28. A) Find
    13·1 answer
  • [25 - (2 + 6) + 3] / 2
    12·1 answer
  • What is the value of x for the point with a y-value of 6?
    14·2 answers
  • David bought a poster for an art project. The poster is 2.7 ft wide and 3.9 ft tall. What is the area of the poster? Enter your
    8·2 answers
  • your team wins 18 medals at a track meet the medals are gold silver and bronze in ratio of 2;2;5. how many of each medal were wo
    10·1 answer
  • What is 3 to the power of 4 simpifyed
    10·2 answers
  • A baseball is thrown upwards from a height of 5 feet with an initial speed of 64 feet per second, and its height h (in feet) fro
    13·3 answers
  • First answer gets brainliest
    13·1 answer
  • How do i solve reflections
    5·1 answer
  • Solve this question with full working and explanation and I will mark you as brainliest. ​
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!