Taking a cutting from a plant is also known as propagation.   If you have a hardwood plant that you want to propagate the process is slow but easy to do as the cuttings are quite hardy.  Take the cutting (about 5 inches worth) when the tree is dormant (in the middle of winter) and place the cutting cut-side up in a pail of sand.  Fill the pail with water and wait until spring.  Then submersed side will have little nodules that will turn into roots once planted.

When you are taking a cutting from a soft-wood plant, it will require more care and attention.  Taking cuttings from soft-wood yields quicker results and you do the actually cutting when the plant is in the active growing phase.  You are cutting off the new growth back to a nodal (from the point it stopped growing the year previous).  This method
benefits from the use of a root producing medium found at nursery stores.  It is important to place the cutting (cut-side down) into the root medium immediately.  If the cutting dries out it will lessen the changes of a successful cutting.

Next, pick a pot or container suitable to the type of plant you are growing and plant it once the roots have developed.  At this point, you can follow the regular care instructions for the type of plant you have just propagated.

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How to Clone any plant for hydroponic or regular gardening