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
AleksandrR [38]
2 years ago
12

Differentiate the following with respect to xirrelevant answers will be reported​

Mathematics
1 answer:
ale4655 [162]2 years ago
4 0

Answer:

\displaystyle y' = - \frac{e^{x^2 + 7} \sqrt{\csc 5x} \Bigg[ \bigg[ 5 \cot (5x) - 4x \bigg] \sin (3x + 4) - 6 \cos (3x + 4) \Bigg] }{2}

General Formulas and Concepts:

<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

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

Derivative Property [Addition/Subtraction]:
\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]

Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

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

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

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify.</em>

<em />\displaystyle y = e^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)}

<u>Step 2: Differentiate</u>

  1. Apply Derivative Rule [Product Rule]:
    \displaystyle y' = \big[ e^{x^2 + 7} \big]' \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'
  2. Apply Exponential Differentiation [Derivative Rule - Chain Rule]:
    \displaystyle y' = e^{x^2 + 7} (x^2 + 7)' \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'
  3. Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \big[ \sin (3x + 4) \big]' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'
  4. Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \cos (3x + 4) (3x + 4)' \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'
  5. Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \big[ \sqrt{\csc (5x)} \big]'
  6. Apply Derivative Rules [Basic Power Rule + Chain Rule]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \frac{\big[ \csc (5x) \big] '}{2\sqrt{\csc (5x)}}
  7. Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \frac{- \csc (5x) \cot (5x) (5x)'}{2\sqrt{\csc (5x)}}
  8. Apply Derivative Rules and Properties [Basic Power Rule + Multiplied Constant]:
    \displaystyle y' = 2xe^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)} + 3e^{x^2 + 7} \cos (3x + 4) \sqrt{\csc (5x)} + e^{x^2 + 7} \sin (3x + 4) \frac{-5 \csc (5x) \cot (5x)}{2\sqrt{\csc (5x)}}
  9. Rewrite:
    \displaystyle y' = - \frac{e^{x^2 + 7} \sqrt{\csc 5x} \Bigg[ \bigg[ 5 \cot (5x) - 4x \bigg] \sin (3x + 4) - 6 \cos (3x + 4) \Bigg] }{2}

∴ we have found the derivative of the function.

---

Learn more about differentiation: brainly.com/question/26836290
Learn more about calculus: brainly.com/question/23558817

---

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

Unit: Differentiation

You might be interested in
Three values on a number line are labeled F, G, and H . Which number line correctly shows the values of F, G, and H?​
oksian1 [2.3K]

Answer:

C

Step-by-step explanation:

F must be on -4

H must be on 4 (because it is -f=-(-4)=-1•-4=4)

6 0
2 years ago
What is the inverse to 2^x-1
nydimaria [60]

Replace x with y and solve for y~

y=2*

x=2^y

log x = y log 2

y=log x / log 2

Hope this helps and leave a brainliest to help me reach expert ;)

4 0
3 years ago
Can anyone tell me why I can't watch the ads on here to view answers anymore?
GenaCL600 [577]

Answer:

probably an update which sucks tell me about it lol

7 0
3 years ago
Read 2 more answers
A rectangle has a perimeter of 30 inches.
erastova [34]
The new perimeter would be 60. because the width and the height are doubled.
8 0
3 years ago
An academy requires there to be 3 teachers for every 72 students and 5 tutors for every 60 students. How many students does the
vova2212 [387]
For every 24 students there is one teacher. For every 12 students is one tutor. If the Academy had 132 students then you will need 11 tutors.
5 0
2 years ago
Other questions:
  • What is 5(3-12)(42-32)
    9·2 answers
  • Which sentence is a run-on sentence?
    5·2 answers
  • If a family uses 3 gallons of milk every week, how much milk will this family use in four days?
    8·2 answers
  • RJ’s soccer team has a record of winning 2 matches for every 3 they lose. If the team lost 15 matches during the season, how man
    5·1 answer
  • Translate the verbal phrase into an algebraic expression
    7·1 answer
  • Please help, I'm struggling with this so much! (8n^5-3n^4-56n^2+13n)÷(8n-3)
    6·1 answer
  • Which expression is equivalent to 4^6 ⋅ 4^−8?
    7·1 answer
  • Find the least common multiple of these two expressions.<br> 4v^7 w^5 and 10v^2w^4y^8
    13·1 answer
  • The volume of the Jupiter is a 143300000000000 write the stradard form​
    8·1 answer
  • Why is 3 1/2 multipiled by x in the equation
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!