# How do you use the first and second derivatives to sketch #y=e^x/x#?

##### 1 Answer

graph{(e^x)/x [-22, 18, -9.88, 10.12]}

#### Explanation:

to sketch the graph, we need to examine various behaviour of the function

**roots:**

But as

**behaviour as #x->+-oo#**

As

As

**asymptotes**

denominator

so a vertical asymptote when

**turning (or critical points)**

gives:

At a critical point

When

so there is a critical point at

**Nature of the critical points:**

We need to look at the second derivative;

we can rearrange

we already know the derivative of

so we get

That's quite a complex expression, so lets not even bother trying to simplify any more as we increase the chance of making a mistake; just substitute

**Sketching the graph:**

We now have enough to sketch the graph which actually looks like this;

graph{(e^x)/x [-22, 18, -9.88, 10.12]}