Linear plus reciprocal to a fractional power: Find for .

We can use the Chain Rule by defining the quantity in parentheses as a temporary variable u:

Let . Then , and we can apply the Chain Rule: .

We determine the two derivatives separately and then combine results. On the next derivative, we use the Power Rule:

. For we need to convert the reciprocal to a power:

We see that we need to apply the Chain Rule by applying the Power Rule to the quantity and then take the derivative of the quantity:

.

Combining these results, we get

. We can express this entirely in terms of x by using the definition of u above: