Estimate 64,505+38,972
2 answers:
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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)
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(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
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(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)
We are given that a coin is toseed with two possible outcomes (head or tails) and a ball from a ball with balls of three different colors is picked.
We have that the possible outcomes are all possible combinations of the faces of the coin and the color of one of the balls.
Therfore, if:
The possible combinations are:
Therefore, the right answer is C.