What is the instantaneous rate of change of #f(x)=x/(-x-8)# at #x=4 #?

Answer 1

#-1/18#

#"the instantaneous rate of change at x = 4"#
#"is the value of the derivative at x = 4"#
#"differentiate using the "color(blue)"quotient rule"#
#"given "f(x)=(g(x))/(h(x))" then"#
#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#
#g(x)=xrArrg'(x)=1#
#h(x)=-x-8rArrh'(x)=-1#
#rArrf'(x)=(-x-8-x(-1))/(-x-8)^2=-8/(-x-8)^2#
#rArrf'(4)=-8/(144)=-1/18#
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Answer 2

# \ #

# "Answer is:" \qquad \quad - 1/18. #

# \ #
# "The required quantity is the derivative of" \ \ f(x) \ \ "evaluated at" # # \qquad \qquad x = 4. #
# "So, let's calculate the derivative of" \ \ f(x): #
# :. "instantaneous rate of change of"\ \ f(x) \ \ "at" \ (x=4) \ = - 1/18.#
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Answer 3

To find the instantaneous rate of change of (f(x) = \frac{x}{-x-8}) at (x = 4), we'll first find the derivative of the function (f(x)) with respect to (x), and then evaluate it at (x = 4).

Given (f(x) = \frac{x}{-x-8}), we'll use the quotient rule to find its derivative:

[f'(x) = \frac{d}{dx}\left(\frac{x}{-x-8}\right)]

Using the quotient rule:

[f'(x) = \frac{(1)(-x - 8) - (x)(-1)}{(-x - 8)^2}]

[f'(x) = \frac{-x - 8 + x}{(-x - 8)^2}]

[f'(x) = \frac{-8}{(-x - 8)^2}]

Now, we'll evaluate the derivative (f'(x)) at (x = 4):

[f'(4) = \frac{-8}{(-4 - 8)^2}]

[f'(4) = \frac{-8}{(-12)^2}]

[f'(4) = \frac{-8}{144}]

[f'(4) = -\frac{1}{18}]

So, the instantaneous rate of change of (f(x)) at (x = 4) is (-\frac{1}{18}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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