differentiate the following with respect to x
irrelevant answers will be reported
Answers
Answer:
[tex]\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}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]:
[tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]:
[tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]:
[tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle y = e^{x^2 + 7} \sin (3x + 4) \sqrt{\csc (5x)}[/tex]
Step 2: Differentiate
Apply Derivative Rule [Product Rule]:[tex]\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]'[/tex]Apply Exponential Differentiation [Derivative Rule - Chain Rule]:
[tex]\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]'[/tex]Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
[tex]\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]'[/tex]Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:
[tex]\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]'[/tex]Apply Derivative Rules and Properties [Basic Power Rule + Addition/Subtraction]:
[tex]\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]'[/tex]Apply Derivative Rules [Basic Power Rule + Chain Rule]:
[tex]\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)}}[/tex]Apply Trigonometric Differentiation [Derivative Rule - Chain Rule]:
[tex]\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)}}[/tex]Apply Derivative Rules and Properties [Basic Power Rule + Multiplied Constant]:
[tex]\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)}}[/tex]Rewrite:
[tex]\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}[/tex]
∴ we have found the derivative of the function.
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Learn more about differentiation: https://brainly.com/question/26836290
Learn more about calculus: https://brainly.com/question/23558817
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
To find the derivative of the function [tex]\displaystyle\sf\:y=e^{x^{2}+7}\sin(3x+4)\sqrt{\csc(5x)}}[/tex] with respect to [tex]\displaystyle\sf x[/tex], we'll use the chain rule and product rule.
Let's break down the function into its individual parts:
[tex]\displaystyle\sf u= e^{x^{2}+7}[/tex]
[tex]\displaystyle\sf v= \sin(3x+4)[/tex]
[tex]\displaystyle\sf w= \sqrt{\csc(5x)}}[/tex]
Now, let's differentiate each part separately:
Using the chain rule, the derivative of [tex]\displaystyle\sf u= e^{x^{2}+7}[/tex] is:
[tex]\displaystyle\sf \dfrac{du}{dx}= (2x)e^{x^{2}+7}[/tex]
Using the chain rule, the derivative of [tex]\displaystyle\sf v= \sin(3x+4)[/tex] is:
[tex]\displaystyle\sf \dfrac{dv}{dx}= 3\cos(3x+4)[/tex]
To differentiate [tex]\displaystyle\sf w= \sqrt{\csc(5x)}}[/tex], we can rewrite it as [tex]\displaystyle\sf w= (\csc(5x))^{1/2}[/tex]. Using the chain rule, the derivative of [tex]\displaystyle\sf w[/tex] is:
[tex]\displaystyle\sf \dfrac{dw}{dx}= \dfrac{1}{2}(\csc(5x))^{-1/2}(-\cot(5x))(5)[/tex]
Simplifying the derivative [tex]\displaystyle\sf \dfrac{dw}{dx}[/tex]:
[tex]\displaystyle\sf \dfrac{dw}{dx}= -\dfrac{5\cot(5x)}{2\sqrt{\csc(5x)}}[/tex]
Now, using the product rule, the derivative of [tex]\displaystyle\sf y[/tex] with respect to [tex]\displaystyle\sf x[/tex] can be calculated as follows:
[tex]\displaystyle\sf \dfrac{dy}{dx}= \dfrac{du}{dx} \cdot v \cdot w + u \cdot \dfrac{dv}{dx} \cdot w + u \cdot v \cdot \dfrac{dw}{dx}[/tex]
Substituting the values of [tex]\displaystyle\sf \dfrac{du}{dx}[/tex], [tex]\displaystyle\sf \dfrac{dv}{dx}[/tex], and [tex]\displaystyle\sf \dfrac{dw}{dx}[/tex]:
[tex]\displaystyle\sf \dfrac{dy}{dx}= \left((2x)e^{x^{2}+7}\right) \cdot \sin(3x+4) \cdot \sqrt{\csc(5x)} + e^{x^{2}+7} \cdot \left(3\cos(3x+4)\right) \cdot \sqrt{\csc(5x)} - e^{x^{2}+7} \cdot \sin(3x+4) \cdot \dfrac{5\cot(5x)}{2\sqrt{\csc(5x)}}[/tex]
Simplifying further:
[tex]\displaystyle\sf \dfrac{dy}{dx}= \left(2xe^{x^{2}+7}\right) \sin(3x+4) \sqrt{\csc(5x)} + 3e^{x^{2}+7}\cos(3x+4) \sqrt{\csc(5x)} - \dfrac{5e^{x^{2}+7} \sin(3x+4) \cot(5x)}{2\sqrt{\csc(5x)}}[/tex]
Therefore, the derivative of [tex]\displaystyle\sf y[/tex] with respect to [tex]\displaystyle\sf x[/tex] is:
[tex]\displaystyle\sf y' = -2e^{x^{2}+7}\csc(5x)[5\cot(5x)-4x]\sin(3x+4) - 3e^{x^{2}+7}\sqrt{\csc(5x)}\cos(3x+4) - \dfrac{5e^{x^{2}+7}\sin(3x+4)\cot(5x)}{2\sqrt{\csc(5x)}}[/tex]
Simplifying further, we have:
[tex]\displaystyle\sf y' = -2e^{x^{2}+7}\csc(5x)[5\cot(5x)-4x \sin(3x+4) - 6e^{x^{2}+7}\cos(3x+4)\sqrt{\csc(5x)}[/tex]
Therefore, the derivative of [tex]\displaystyle\sf y[/tex] with respect to [tex]\displaystyle\sf x[/tex] is:
[tex]\displaystyle\sf y' = -2e^{x^{2}+7}\csc(5x)[5\cot(5x)-4x]\sin(3x+4) - 6e^{x^{2}+7}\cos(3x+4)\sqrt{\csc(5x)}[/tex]
[tex]\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}[/tex]
♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]
Related Questions
y = x + 2
5x - 4y = -3
Solving systems by equations
Answers
Answer: x = 5, y = 7
Step-by-step explanation:
First, put the variables together/rearrange:
y - x = 2 & 5x - 4y = -3
Cancel out variable x by multiply the first equation by 5:
5y - 5x = 10 & 5x - 4y = -3
Add equations:
y = 7
Then determine x by plugging y back into the first equation y = x + 2:
x = 5
Write 1.5 as an improper fraction and mixed number in simplest form.
Answers
Answer:
1 1/2 (mixed)
3/2 (improper)
Step-by-step explanation:
When the one is put to the side, you get 5/10 if you turn "0.5" into a frsaction. if you simplify 5/10, you get 1/2. Therefore, 1 1/2 is your answer. Your not done yet! After, you need to change 1 1/2 into a improper fraction which is 3/2.
good luck :)
3/2
is the answer to this question
Three adjacent angles form a right angle . The measure of the largest angle is 5 degrees less than 3 times the measure of the smallest angle The measure of the median angle is 5 degrees more than the measure of the smallest angle . What is the measure of the largest angle ?
Answers
Answer:
Step-by-step explanation:
Amy and Tina each by a coffee.
Amy gets 15p change from £2
Tina gets £2.45 change from £5
How much more does Tina pay than Amy?
Give your answer in pence.
Answers
Answer:
70p
Step-by-step explanation:
Amy-
£2.00 - £0.15 = £1.85
so Amy payed £1.85
Tina-
£5.00 - £2.45 = £2.55
so Tina payed £2.55
£2.55 - £1.85 = £0.70
which means tina payed 70p more
Find the total amount
$16.000 at 14% compounded
semiannually for 2 years
Answers
Answer:
The answer would be 20.97273616 or (rounded to the nearest cent) would be $20.97
Step-by-step explanation:
The reason for this is that the formula for compounded interest is [tex]A=P(1+\frac{r}{n})^\frac{t}{n}[/tex] where A is the total amount, P is the initial amount, r is the growth factor or the percent, n is the number of compounding periods, and t is the time. You would substitute P for 16, r for 0.14 (14%), n for 2 as it compounded semiannually so it for every 6 months instead of 1 year (annual) so is compounded 2 times per year, and t for 2. And just solve the equation and you get that A is equal to 20.97273616 but since we are talking about money( only round if asked) then it would be 20.97.
Hope this helps!!!!
Sarah answered 70% of the trivia questions correctly. What fraction describes this percent? Enter your answer as a simplified fraction The fraction that describes this percent is
Answers
Answer:
7/10
Step-by-step explanation:
70 out of 100 as a fraction would be 70/100 but since they said simplify it becomes 7/10.
This cannot be made smaller, as 7 cannot be divided into 2 evenly.
Hope this helps!
if x=-2 then write the ordered pair that is a solution of 5x-3y=-25
Answers
Answer:
(2,11 2/3) or (2,(35/3))
Step-by-step explanation:
5x-3y=-25
For x = 2:
5(2)-3y=-25
10-3y=-25
-3y=-35
y = (35/3) or 11 2/3
(2,11 2/3)
Answer:
X = -2
Y= 5
You use:
5x-3(5)=-25
and
5(-2)-3y=-25
!Need answer quickly!
The preimage of the triangle (yellow) has the following coordinates:
A. (3,-4) (red dot)
B. (9,0) (yellow dot)
C. (7,-6) (blue dot)
Describe the translation from the preimage to the image and include how the x and y values of the ordered pairs changed. (Please put real answer or else I'll report you)
Answers
Answer:
(x-5;y+6)
Step-by-step explanation:
1) the difference between the coordinates before translation and after the translation is*: Δx=-5; Δy=+6, then
2) the rule of the translation is: (x-5;y+6).
* for example, the red dot. The difference for X is: -2-3=-5. The difference for Y is: 2--4=+6.
Angela had 8 computer games.
She got 3 more for her birthday. But then she gave
away 4 computer games.
4
How many computer games did Angela have then?
Answers
[tex]\huge\underline{\red{A}\blue{n}\pink{s}\purple{w}\orange{e}\green{r} -}[/tex]
we're given that Angela formerly had 8 computer games.
★ She got 3 more video games on her birthday.
★ Then she gave away 4 video games.
_____________________________
To calculate - number of computer games she's finally left with._____________________________
[tex]\huge\red{ Solution -}[/tex]
[tex]8 + 3 - 4 \\ \dashrightarrow \: 11 - 4 \\ \dashrightarrow \: 7[/tex]
hope helpful :D
help pls first correct answer gets brainleist
Answers
Answer:
18
Step-by-step explanation:
a^2 + b^2 = c^2
in this case 24 is a and 30 is c
24^2 = 576
30^2 = 900
900 - 576 = 324
the square root of 324 is 18
What is a number that is 450 rounded to the nearest 10th
What is a number that is 400when rounded to the nearest hundred
Answers
Step-by-step explanation:
e.g. 452 and 397
but also e.g. 448 and 429
If you know the diameter of a circle,
how do you find the following. Explain
the steps clearly
a. the area of that circle?
b. the circumference of the circle?
Answers
Step-by-step explanation:
a. Area of a circle = πr², where r means radius.
Step 1: converted diameter to radius.
diameter = 2 radius
In other way of writing, radius = [tex]\frac{1}{2}[/tex] diameter.
Step 2: substitute radius into the formula.
Area of a circle = π([tex]\frac{1}{2}[/tex]d)²
Step 3: substitute π into the formula.
some question is given the value of π), if not substitute 3.14 for π.
Area of a circle = 3.14([tex]\frac{1}{2}[/tex]d)²
Step 4: substitute d into the formula, then we can solve its area.
b. a. Circumference of a circle = 2πr, where r means radius.
Step 1: converted diameter to radius.
diameter = 2 radius
In other way of writing, radius = [tex]\frac{1}{2}[/tex] diameter.
Step 2: substitute radius into the formula.
C = 2π([tex]\frac{1}{2}[/tex]d)
Step 3: substitute π into the formula.
some question is given the value of π), if not substitute 3.14 for π.
C = 2(3.14)([tex]\frac{1}{2}[/tex]d)
Step 4: substitute d into the formula, then we can solve its circumference.
(PLEASE HELP)(THIS IS MIDDLE SCHOOL LEVEL EASY!) What is the surface area of the box
Answers
Answer:
C) 42 square cm
Step-by-step explanation:
each rectangle 2x3 = 6
each square 3x3 =9
4 rectangles 4x6 = 24
2 squares 2x9=18
24+18 = 42
can someone help me please. I’ll give you ratings
Answers
Answer:
(c) -0.9682
Step-by-step explanation:
In quadrant II, the cosine is negative. Solving the given equation for the cosine gives ...
sin²(A) +cos²(A) = 1
cos²(A) = 1 -sin²(A)
cos(A) = -√(1 -sin²(A)) = -√(1 -(1/4)²) = -√(15/16) ≈ -0.9682
_____
The coordinates of a point on the unit circle are (cos(α), sin(α)). The attachment shows the point on the unit circle where sin(A)=1/4 in quadrant II. The x-coordinate is the cosine of the angle, about -0.97.
Candice is putting together a business outfit. There are 6 pairs of pants and 2 dress shirts to choose from. How many different outfits can Candice put together?
Answers
a customer wants you to enlarge a photo to 2 3/4 its current height. the photos current height is 3 1/4 inches. what should its enlarged height be in inches?
Answers
this a a joke...
haha goodbye
Answer:6 inches
Step-by-step explanation:
The diagonals of rhombus $ABCD$ intersect at $E$ . Given that $m\angle BAC=53°$ , $DE=8$ , and $EC=6$ , find $m\angle AED$ .
Answers
Based on the definition of the diagonals of a rhombus, m∠AED = 90°.
What are the Diagonals of a Rhombus?A rhombus has two diagonals that bisect each other at right angles, that is they are perpendicular to each other.
Angle 90 is formed at the intersection of the two diagonals of a rhombus.
In rhombus ABCD, ∠AED is formed at the intersection of the diagonals of the rhombus.
Therefore, based on the definition of the diagonals of a rhombus, m∠AED = 90°.
Learn more about diagonals of a rhombus on:
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Pleasee help it’s due on Monday
Answers
Answer:
1.6 × [tex]10^{10}[/tex]
Step-by-step explanation:
( 4 × [tex]10^{5}[/tex] )( 4 × [tex]10^{4}[/tex] ) = ( 4 · 4 ) × ( [tex]10^{5}[/tex] · [tex]10^{4}[/tex] ) = 16 × [tex]10^{4+5}[/tex] = ( 1.6 × [tex]10^{1}[/tex] ) × [tex]10^{9}[/tex] = 1.6 × [tex]10^{10}[/tex]
What is the volume of a hemisphere with a radius of 2.3 m, rounded to the nearest tenth of a cubic meter?
Answers
Answer:
25.3m³
Step-by-step explanation:
To find how high school students feel about hot lunches, adam walks to the nearest high school and gives a survey postcard to every twentieth student that enters the building. he asks the students to mail the postcard in if they choose to participate in the study. why might the method used be biased?
Answers
The statement second and statment third are the reason that method used might be biased.
It is given that Adam gives a survey postcard to every twentieth student that enters the building.
It is required to find the statement why the method used to be biased.
What is the sample size?The sample size is nothing but the number of observations in an activity or participants generally it is represented by the small letter 'n'.
Adam gives a survey postcard to every twentieth student, not every student that enters the building.
It means asking every twentieth student does not represent the student population.
Also, the opinions of students at one high school do not exactly reflect the opinions of students at other high schools.
Students that take the time to fill out the survey and send it in are likely passionate about hot lunches.
Thus, the statement second and statment third are the reason that method used might be biased.
Learn more about the biased sample here:
https://brainly.com/question/26962814
Answer:
B and C
Step-by-step explanation:
What is the volume of this figure?
Enter your answer in the box.
ft³
Three-dimensional figure that could be formed by placing two rectangular prisms together to form an L shape where the wider part of the L is on the bottom and the L extends up from the left side. The wider part of the L has a length of 6 ft and a width of 2 ft and a height of 3 ft. The L extends up on the left side with the total height of the figure labeled 8 ft. The taller part of the L has a length of 4 ft and a width of 2 ft.
Answers
The volume of the L-shaped figure would be the total volume of the two rectangular prisms that forms the shape.
What is the Volume of a Rectangular Prism?The volume of a rectangular prism = length × width × height
The image referred to in this question is missing, however, to find the volume of the figure that has L-shape, decompose the figure into two rectangular prism and find the volume of each rectangular prism.
The volume of the figure would be the total volume of the two rectangular prisms.
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Can someone please help me ASAP
Answers
What is the x to help me
Answers
Answer:
116 degrees
Step-by-step explanation:
the shape is 360 degrees in total
64+64 = 128
360-128 = 232
232/2 = 116
Is my answer correct?
Answers
Answer:
No, x = 3 inStep-by-step explanation:
The area of the rhombus is half the product of its diagonals
A = 1/2(2x*16) = 4816x = 48x = 48/16x = 3[tex]area \: of \: rectangle \\= 4 \times area \: of \:identical \: triangle[/tex]
Explanation :-[tex]area \: of \: rectangle =\\ 4 \times area \: of \:identical \: triangle[/tex]
[tex]48 {in}^{2} = 4 \times \frac{1}{2} \times base \times height \\ 48 {in}^{2} = 2 \times base \times height \\ \frac{48}{2} = 8 \times height \\ \frac{ \cancel{48} {}^{ \: \: 24} }{ \cancel{2}} = 8 \times height \\ \frac{24}{8} = height \\ \frac{ \cancel{24} {}^{ \: \: 3} }{ \cancel{8}} = height \: \\ height = x = 3in \: [/tex]
The picture shows 3/8 of Puja's penny collection.
*nine pennies*
A. What fraction of her collection is 6 pennies?
B. What fraction of her collection is 21 pennies?
Answers
The fraction of her collection that is 21pennies is 56
Ratio and proportionsGiven the fraction for Puja's collection expressed as 3/8
Leet x be the fraction of her collection which is 6 pennies, to have;
x of 3/8 = 6
3x/8 = 6
3x = 48
x = 16
Hence the fraction of her collection that is 6pennies is 16
Let y be the fraction of her collection which is 21 pennies, to have;
x of 3/8 = 21
3x/8 = 21
x = 56
Hence the fraction of her collection that is 21pennies is 56
Learn more on ratio and proportions here: https://brainly.com/question/19994681
Does anyone know this?
Answers
Answer:
9
Step-by-step explanation:
y= mx+b
m is slope
b is y-int
I need answer this question
Answers
Answer:
5/12
is the correct answer
see my question and come there now females
Step-by-step explanation:
Goal: illustrates radius, diameter, chord, center, arc, central angle, and
inscribed angle.
Role: Garden Designer/Landscape Architect
Audience: Math Teacher
Situation: A certified plantita and backyard owner plans to have a unique
garden. As a Landscape Architect, you are asked to make a sample layout of
a circular garden. She asks you to include the following on her garden:
● well
● pond
● bridge
● fountain
● walkways
● benches or chairs
● entrance and exit gates.
● trees, ornamental plants, and herbs
Procedure:
1. Draw a circle.
2. Conceptualize a circular garden and layout using the specified
items mentioned above.
What I Can Do
9
DISCIPLINE • GOOD TASTE • EXCELLENCE
3. Label the parts of circle formed on the garden layout
Please I'm looking for a serious answer. I need to pass this on monday. Thank you.
Which symbol makes the comparisons true ? Complete the comparison -6 ? 3
Answers
[tex]======================================[/tex]
Negative numbers are always less than positive ones.
Do you remember the symbol for "less than"? That's right, "<" :)
Hence,
-6<3
Because
-6 is negative (the - sign tells us that)
3 is positive
[tex]===================================[/tex]
Notes: Hope everything is clear.Let me know if you have any questions!Answered by:~DiamondS~
what is the answer to
-40 = -5p
Answers
Answer:
-200 =p
Step-by-step explanation:
-40 = -5p
5 5
_______
-200 = p
Sketch the graph of y=x^2-4x+3
Answers
Answer:
see attached graph
Step-by-step explanation:
Given equation: [tex]y=x^2-4x+3[/tex]
Standard form of a quadratic equation: [tex]y=ax^2+bx+c[/tex]
where [tex]c[/tex] is the y-intercept
If [tex]a > 0[/tex] then the parabola opens upwards.
If [tex]a < 0[/tex] then the parabola opens downwards.
Therefore, the y-intercept of the graph is (0, 3)
To find the x-intercepts, factor the equation:
[tex]y=x^2-4x+3[/tex]
[tex]\implies y=x^2-x-3x+3[/tex]
[tex]\implies y=x(x-1)-3(x-1)[/tex]
[tex]\implies y=(x-3)(x-1)[/tex]
Therefore, the x-intercepts of the graph are (3, 0) and (1, 0)
To determine the vertex, differentiate the equation:
[tex]\implies \dfrac{dy}{dx}=2x-4[/tex]
Set to zero and solve for x:
[tex]2x-4=0 \implies x=2[/tex]
Substitute [tex]x=2[/tex] into the original equation and solve for y:
[tex]\implies y=(2)^2-4(2)+3=-1[/tex]
Therefore, the vertex (or turning point) is at (2, -1)