This week we learned implicit derivatives. Implicit derivatives are very easy if you know your derivative rules. Implicit derivatives are just regular derivatives with more than one variable. What this means is you derive it with respect to another variable. After you derive with your regular derivative rules, the only difference is the step afterwards. When you derive the second variable you have to show it is in respect to that variable, so you take your derivative and multiply by dy/dx or dt/dx etc. After deriving, you solver for your dy/dx and that is your implicit derivative. Now this may seem complicating in words, but lets see a few examples.
x^2+y^2=10
First step is derive the equation using the normal derivative rules.
2x+2y(dy/dx)=0
I added the dy/dx to show that the derivative is in respect to y. The 10 became a 0 because you must derive both sides of the equation just like anything in algebra and due to the constant rule the derivative of 10 is zero. The next step is to solve for dy/dx using basic algebra.
2y(dy/dx)=-2x
dy/dx=-2x/2y
dy/dx=-x/y
Therefore -x/y is your derivative. If a point was given you could plug in at this time. If it asked for a second derivative of this equation though, the steps would be quite different. You take your first derivative and start to derive it like normal, but you plug in your first derivative where it occurs. And i guess thats about it about implicit derivatives. They do have word problems, that require some thinking outside of the box, but i don't have my book for examples and I really don't feel like making up a word problem at the moment. Good night.
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