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How do you find the instantaneous slope of #y=1/x# at x=4?

Answer 1

Emily Ammerman

#-1/16#

The instantaneous slope is determined by differentiating #y# at #x=4#

Differentiating #y# is computed by applying the quotient rule #color(blue)((u/v)'=(u'v-v'u)/v^2#

#y=1/x#

#y'=color(blue)((1'xxx-x'xx1)/x^2)#

#y'=0-1/x^2#

#y'=-1/x^2# #y'_((x=4))=-1/4^2=-1/16#

Therefore, the slope of y at #x=4# is #-1/16#

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Answer 2

Scarlett Allison

To find the instantaneous slope of ( y = \frac{1}{x} ) at ( x = 4 ), you need to calculate the derivative of the function at that point. Using the power rule for differentiation, the derivative of ( \frac{1}{x} ) is ( -\frac{1}{x^2} ). Substituting ( x = 4 ) into the derivative, you get ( -\frac{1}{4^2} = -\frac{1}{16} ). Therefore, the instantaneous slope of ( y = \frac{1}{x} ) at ( x = 4 ) is ( -\frac{1}{16} ).

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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.