INPUT ARTICLE: Article: The first rule you probably learned for finding derivatives is the power rule. This rule says that for a variable x{\displaystyle x} raised to any exponent a{\displaystyle a}, the derivative is as follows: f(x)=xa{\displaystyle f(x)=x^{a}} f(x)=axa1{\displaystyle f^{\prime }(x)=ax^{a-1}} For example, review the following functions and their derivatives: If f(x)=x2{\displaystyle f(x)=x^{2}}, then f(x)=2x{\displaystyle f^{\prime }(x)=2x} If f(x)=3x2{\displaystyle f(x)=3x^{2}}, then f(x)=23x=6x{\displaystyle f^{\prime }(x)=2*3x=6x} If f(x)=x3{\displaystyle f(x)=x^{3}}, then f(x)=3x2{\displaystyle f^{\prime }(x)=3x^{2}} If f(x)=12x4{\displaystyle f(x)={
rac {1}{2}}x^{4}}, then f(x)=412x3=2x3{\displaystyle f^{\prime }(x)=4*{
rac {1}{2}}x^{3}=2x^{3}} To find the derivative of a square root function, you need to remember that the square root of any number or variable can also be written as an exponent. The term below the square root (radical) sign is written as the base, and it is raised to the exponent of 1/2. Consider the following examples: x=x12{\displaystyle {\sqrt {x}}=x^{
rac {1}{2}}} 4=412{\displaystyle {\sqrt {4}}=4^{
rac {1}{2}}} 3x=(3x)12{\displaystyle {\sqrt {3x}}=(3x)^{
rac {1}{2}}} If the function is the simplest square root, f(x)=x{\displaystyle f(x)={\sqrt {x}}}, apply the power rule as follows to find the derivative: f(x)=x{\displaystyle f(x)={\sqrt {x}}\ \ \ \ \ }(Write the original function.) f(x)=x(12){\displaystyle f(x)=x^{({
rac {1}{2}})}\ \ \ \ \ }(Rewrite the radical as an exponent.) f(x)=12x(121){\displaystyle f^{\prime }(x)={
rac {1}{2}}x^{({
rac {1}{2}}-1)}\ \ \ }(Find derivative with the power rule.) f(x)=12x(12){\displaystyle f^{\prime }(x)={
rac {1}{2}}x^{(-{
rac {1}{2}})}\ \ \ }(Simplify exponent.) At this stage, you need to recognize that a negative exponent means to take the reciprocal of what the number would be with the positive exponent. The exponent of 12{\displaystyle -{
rac {1}{2}}} means that you will have the square root of the base as the denominator of a fraction. Continuing with the square root of x function from above, the derivative can be simplified as: f(x)=12x12{\displaystyle f^{\prime }(x)={
rac {1}{2}}x^{-{
rac {1}{2}}}} f(x)=121x{\displaystyle f^{\prime }(x)={
rac {1}{2}}*{
rac {1}{\sqrt {x}}}} f(x)=12x{\displaystyle f^{\prime }(x)={
rac {1}{2{\sqrt {x}}}}}

SUMMARY: Review the power rule for derivatives. Rewrite the square root as an exponent. Apply the power rule. Simplify the result.

INPUT ARTICLE: Article: Keep your cleaver and your chef's knife in the drawer for this job. To properly bone a chicken, it helps to have a knife with a fairly flexible, but quite sharp, blade that you can use to work alongside the bones and remove them, as well as scrape the meat from the more difficult joints. Find the backbone. You should be able to feel quite easily where the backbone is with your finger, and carefully work the flat edge of your boning knife down the back, to one side of the bone. Use the bone as a guide, and start working your knife into the skin to create a starting point. You may find it helpful to cut the skin in several places, then flip the knife edge up and cut through the skin from underneath. You may also find it easier to cut just to the left or the right of the spine when you get started. Grasp the skin with one hand, and (carefully!) cut the meat away from the bone, pulling it free as you work. Start by grasping the skin next to the spine cut furthest away from you. Cut as close to the bone as you can, letting your knife do the work. As you start pulling the chicken away from the ribs, you'll encounter the wishbone quickly. Spin the chicken around so the neck hole is facing you, and work your knife around the wishbone to loosen it, pulling it free. The wishbone is famously easy to break, and might snap as you try to get it out. That's fine, but make sure you get the pieces and be careful of bone shards as you work. Keep cutting meat away from the rib cage, slowly moving down from the back, around the side, toward the breast. You'll feel the wing and leg joints as you move, which you'll need to take a little more care to separate and, later, remove. Work slowly and use pressure to pull the meat away from the ribs to make the knife secondary. Use as small cuts as you can get away with, taking care to not cut through the skin on the other side (the breast). Keep separating until you reach the leg and wing joints. Start working down the other side of the backbone, working your knife around much as before, stopping to separate the leg and wing joints before continuing. Alternatively, you can move to the next step and separate the leg and wing joints before you start on the other side. Save the complete removal of the bones, however, until you've gotten the rib cage completely separated from the meat and can more easily cut them out.

SUMMARY:
Make sure you use a very sharp boning knife. Place the bird breast side down on the cutting board. Begin working your knife down one side of the rib cage. Remove the wishbone. Continue cutting, and find the wing and leg bone on that side. Flip the carcass around and do the same thing.