For what value of #x# is the slope of the tangent line to #y=x^7 + (3/x)# undefined?

Answer 1

Please see below.

Quick answer

The derivative is undefined at #x = 0#
#y' = 7x^6-3/x^2# is defined for all #x# except #0#.

Additional detail

Since #y# is also not defined when #x = 0# there is no tangent line where #x = 0#. So it feels a bot odd to say that the slope of the (non-existent) tangent line is undefined,
Here is the graph of #y = x^7+3/x# graph{y = (x^7)+(3/x) [-14.71, 13.76, -7.89, 6.35]}

Here is a similar graph that is easier to see.

graph{x^3+1/(4x) [-7.35, 6.694, -2.82, 4.197]}

By contrast #y = root(3)x# is defined at #x = 0# but

#y' - 1/(3root(3)x^2) is not defined.

The tangent line at #x = 0# is a vertical line.

graph{x^(1/3) [-4.028, 3.767, -2.224, 1.673]}

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Answer 2

The slope of the tangent line to a curve is undefined where the derivative of the function is undefined or where the derivative is zero in the case of a vertical tangent.

For the function ( y = x^7 + \frac{3}{x} ), the derivative can be found using the power rule and the derivative of (\frac{1}{x}):

[ \frac{dy}{dx} = 7x^6 - 3x^{-2} ]

The derivative is undefined when the denominator is zero, as division by zero is undefined. So, the derivative is undefined when ( x = 0 ). Therefore, the slope of the tangent line to the curve is undefined at ( x = 0 ).

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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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