Can you help me plz
1 answer:
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Let be the amount of food needed for one man for one day. At the beginning, we have enough food for 400 men for 40 days, i.e.
After 10 days, the 400 men will have consumed
food, impliying that the food remaining is
If 200 more men join the garrison, there are now 600 men. Each of them requires food each day, and there are
units of food remaining. They will last
days.
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
= 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) × (x + 4) ← chain rule
= 2(x + 4) × 1
= 2(x + 4)
Then
= (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 × (x² - 1) ← chain rule
= 3(x² - 1)² × 2x
= 6x(x² - 1)²
Then
= 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
= (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 - 3x² + 2x
--------------------------------------------------------------------
(d)
y = 3x³(x² + 4)²
f(x) = 3x³ ⇒ f'(x) = 9x²
g(x) = (x² + 4)²
g'(x) = 2(x² + 4) × (x² + 4) ← chain rule
= 2(x² + 4) × 2x
= 4x(x² + 4)
Then
= 3x³. 4x(x² + 4) + (x² + 4)². 9x²
= 12(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)