Question

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

Answer 1

Please see below.

Explanation:

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]}