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

1 Answer
Aug 18, 2017

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]}