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]}