# What is the instantaneous rate of change of #f(x)=x-e^(x^2-7) # at #x=3 #?

The initial derivative is

Blessings...I hope this clarification is helpful.

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To find the instantaneous rate of change of the function ( f(x) = x - e^{x^2 - 7} ) at ( x = 3 ), we need to calculate the derivative of the function and then evaluate it at ( x = 3 ).

( f'(x) = 1 - \left(2x e^{x^2 - 7}\right) )

Plugging in ( x = 3 ):

( f'(3) = 1 - \left(2 \times 3 \times e^{3^2 - 7}\right) )

( f'(3) = 1 - (6e^2) )

So, the instantaneous rate of change of ( f(x) ) at ( x = 3 ) is ( 1 - 6e^2 ).

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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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