A line connecting a vertex of a scalene triangle with the midpoint of the opposite side is the

Help. Please its urgent show workings.

laiz [17]

Answer:

see explanation

Step-by-step explanation:

There are 2 possible approaches to differentiating these.

Expand the factors and differentiate term by term, or

Use the product rule for differentiation.

I feel they are looking for use of product rule.

Given

y = f(x). g(x) , then

$\frac{dy}{dx}$ = f(x).g'(x) + g(x).f'(x) ← product rule

(a)

y = (2x - 1)(x + 4)²

f(x) = 2x - 1 ⇒ f'(x) = 2

g(x) = (x + 4)²

g'(x) = 2(x + 4) × $\frac{d}{dx}$ (x + 4) ← chain rule

= 2(x + 4) × 1

= 2(x + 4)

Then

$\frac{dy}{dx}$ = (2x - 1). 2(x + 4) + (x + 4)². 2

= 2(2x - 1)(x + 4) + 2(x + 4)² ← factor out 2(x + 4) from each term

= 2(x + 4) (2x - 1 + x + 4)

= 2(x + 4)(3x + 3) ← factor out 3

= 6(x + 4)(x + 1)

--------------------------------------------------------------------------

(b)

y = x(x² - 1)³

f(x) = x ⇒ f'(x) = 1

g(x) = (x² - 1)³

g'(x) = 3(x² - 1)² × $\frac{d}{dx}$ (x² - 1) ← chain rule

= 3(x² - 1)² × 2x

= 6x(x² - 1)²

Then

$\frac{dy}{dx}$ = x. 6x(x² - 1)² + (x² - 1)³. 1

= 6x²(x² - 1)² + (x² - 1)³ ← factor out (x² - 1)²

= (x² - 1)² (6x² + x² - 1)

= (x² - 1)²(7x² - 1)

----------------------------------------------------------------------

(c)

y = (x² - 1)(x³ + 1)

f(x) = x² - 1 ⇒ f'(x) = 2x

g(x) = (x³ + 1) ⇒ g'(x) = 3x²

Then

$\frac{dy}{dx}$ = (x² - 1). 3x² + (x³ + 1), 2x

= 3x²(x² - 1) + 2x(x³ + 1) ← factor out x

= x[3x(x² - 1) + 2(x³ + 1) ]

= x(3x³ - 3x + 2x³ + 2)

= x(5x³ - 3x + 2) ← distribute

= 5 $x^{4}$ - 3x² + 2x

--------------------------------------------------------------------

(d)

y = 3x³(x² + 4)²

f(x) = 3x³ ⇒ f'(x) = 9x²

g(x) = (x² + 4)²

g'(x) = 2(x² + 4) × $\frac{d}{dx}$ (x² + 4) ← chain rule

= 2(x² + 4) × 2x

= 4x(x² + 4)

Then

$\frac{dy}{dx}$ = 3x³. 4x(x² + 4) + (x² + 4)². 9x²

= 12 $x^{4}$ (x² + 4) + 9x²(x² + 4)² ← factor out 3x²(x² + 4)

= 3x²(x² + 4) [ 4x² + 3(x² + 4) ]

= 3x²(x² + 4)(4x² + 3x² + 12)

= 3x²(x² + 4)(7x² + 12)

5 0

2 years ago

A test is worth a total of 150 points and contains 70 questions. Some of the questions are worth 2 points and some are worth

FrozenT [24]

Answer:

65 two-point questions, and 5 four-point questions.

Emma answered 2 four-point questions.

Step-by-step explanation:

1. Assigning variables and writing equations

We will set the number of 2 point questions as "x" and number of 4 point questions as "y"

We know that:

x+y=70 (number of 2 point questions+number of 4 point questions=total number of questions)

2x+4y=150 (number of points that a two point question is worth*number of two point questions+number of points that a four point question is worth*number of four point questions=total points on test)

2. Combining equations

x+y=70, 2x+4y=150

We will solve one of these equations.

x+y=70

x=70-y

Next, we will substitute the other equation with this solution.

2*(70-y)+4y=150

140-2y+4y=150

2y=10

y=5

Use this solution to find x.

x+y=70

x+5=70

x=65

We found that there are 65 two-point questions, and 5 four-point questions. 

3. Finding Emma's test results

In this case, we know how many two-point questions she answered. So, we will only be using one variable in this equation, which is y.

46*2+4y=100

92+4y=100

4y=8

y=2

So, she answered 2 questions correctly. 

4 0

3 years ago

The number of trees planted by the alpine Nursery in April was three more than twice the number of trees planted by the nursery

Readme [11.4K]

Answer:

There were 34 trees planted in March

Step-by-step explanation:

* Lats explain how to solve the problem

- The number of trees planted by the alpine Nursery in April was

three more than twice the number of trees planted by the nursery

in March

- Assume that the number of trees planted in March is x

∵ The number of trees in April was three more than twice the

number of trees on March

∵ The number of trees on March was x

∴ The number of trees in April = 2x + 3

- There were 71 trees planted in April

∵ The number of trees were planted on April was 71

∵ The number of trees in April = 2x + 3

∴ 2x + 3 = 71

- Subtract 3 from both sides

∴ 2x = 68

- Divided both sides by 2

∴ x = 34

∵ x represents the number of trees were planted in March

∴ There were 34 trees planted in March

7 0

2 years ago

Archy [21]

6×4=24 You keep change flip. So you change ÷ to × And flip 1/4 to 4. So it is 6×4=24

5 0

2 years ago

Find the value of x.

LUCKY_DIMON [66]

Answer:

25

Step-by-step explanation:

8 0

2 years ago