# What is the slope of #f(x)=-X/e^(x-x^3) # at #x=-1#?

Slope

We make use of the derivative of quotient for this type of function

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To find the slope of the function ( f(x) = -\frac{x}{e^{x-x^3}} ) at ( x = -1 ), you need to find the derivative of the function and then evaluate it at ( x = -1 ). The derivative of the function with respect to ( x ) is:

( f'(x) = -\frac{e^{x-x^3}(1-x^2)}{(e^{x-x^3})^2} - \frac{xe^{x-x^3}(1-3x^2)}{(e^{x-x^3})^2} )

Now, substitute ( x = -1 ) into ( f'(x) ) to find the slope at ( x = -1 ):

( f'(-1) = -\frac{e^{-1-(-1)^3}(1-(-1)^2)}{(e^{-1-(-1)^3})^2} - \frac{-1e^{-1-(-1)^3}(1-3(-1)^2)}{(e^{-1-(-1)^3})^2} )

( f'(-1) = -\frac{e^{-1+1}(1-1)}{(e^{-1+1})^2} - \frac{-1e^{-1+1}(1-3)}{(e^{-1+1})^2} )

( f'(-1) = -\frac{0}{e^0} - \frac{-2e^0}{e^0} )

( f'(-1) = -\frac{0}{1} - \frac{-2}{1} )

( f'(-1) = 0 + 2 )

( f'(-1) = 2 )

So, the slope of ( f(x) = -\frac{x}{e^{x-x^3}} ) at ( x = -1 ) is ( 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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