What is the instantaneous rate of change of #f(x)=ln(4x^2+2x ) # at #x=-1 #?

Answer 1

#-3#

Instantaneous rate of change is simply the derivative. To find it, take the derivative of the function and evaluate it at the desired #x#-value.
We have a logarithmic function with a polynomial inside, which means we need to use the chain rule. As it applies the the natural log function, the chain rule is: #d/dx(ln(u))=(u')/u# Where #u# is a function of #x#.
In this case, #u=4x^2+2x#, so #u'=8x+2#. Therefore, #f'(x)=(8x+2)/(4x^2+2x)=(2(4x+1))/(2(2x^2+x))=(4x+1)/(2x^2+x)#
All that's left to find instantaneous rate of change is to evaluate this at #x=-1#: #f'(-1)=(4(-1)+1)/(2(-1)^2+(-1))=-3/1=-3#
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Answer 2

To find the instantaneous rate of change of the function ( f(x) = \ln(4x^2 + 2x) ) at ( x = -1 ), we can use the derivative of the function.

The derivative of ( \ln(u) ) is ( \frac{1}{u} \cdot u' ), where ( u ) is the function inside the logarithm.

So, let ( u = 4x^2 + 2x ). Then, ( u' = (8x + 2) ).

Using the chain rule, the derivative of ( f(x) ) is ( \frac{1}{4x^2 + 2x} \cdot (8x + 2) ).

Now, evaluate this derivative at ( x = -1 ):

[ f'(-1) = \frac{1}{4(-1)^2 + 2(-1)} \cdot (8(-1) + 2) ] [ = \frac{1}{4 - 2} \cdot (-8 + 2) ] [ = \frac{1}{2} \cdot (-6) ] [ = -3 ]

So, the instantaneous rate of change of ( f(x) ) at ( x = -1 ) is ( -3 ).

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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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