This video from IntegralCALC shows you how to solve the Reciprocal Rule f(x)=1/(1-2/x) Math problem.
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Reciprocal Rule f(x)=1/(1-2/x) Hi everybody, welcome back. We’re going to be doing for problems with the reciprocal rule which is a basic derivative rule, the rule that helps you take derivatives like product rule, quotient rule, etcetera. So we’re going to do a couple of problems. The first one that we’re going to do f (x) = 1/(1 - 2/x). The reciprocal rule, we’ll just write it down over here. The rule for those of you who don’t still you have it in front of you. Just want to make sure you got it right here. So, basically, all the rules says, that if you’re taking the derivative of function that has 1 in the numerator and anything f(x) in the denominator that the result is going to be in the following form. The derivative of the function on the top, the function square on the bottom with the negative sign in the front. So it’s kind of a shortcut way if you run across something you also take the derivative function like this. You see 1 on the top and a function here on the bottom. It’s a shorthand way to be able to take the derivative using this rule instead of conventionally. So we’re going to go ahead and apply this rule to this function. So we’ll say, f1(x) is going to be, we’ll follow rule so we always need the negative sign here so I’m going to go ahead and write that, negative, and then we need the derivative of the function on the top. So we need to take the derivative of this right here. So we’ll do this term by term, the 1 first and then the 2/x. So the derivative of 1 is zero because one is a constant and then the derivative of 2/x, we can actually change this to be 2/x, I’m going to go ahead and move the x to the top. And the way that I’m going to do that I’ll say -2x-1, this x has an implied, 1 is an exponent on it I’ll move it to the numerator by changing the sign from a positive to a negative on the exponent. So the sign changes to a negative one when we move it to the top. So, the derivative of this one, the way that we do that multiply the exponent by the coefficient so -1 x -2 is simply 2. So we’re going to say 2 and then x and then we subtract 1 from the exponent, always. So -1 – 1 is -2. So I’m going to have -2x-2 and now it’s the derivative of this function right here. So we’ve taken now the derivative and we’ve completed the numerator over here on the rule. So now, what we need to do is deal with the denominator so it’s just simply the function squared. So we’re going to go ahead and write 1 – 2/x and of course we have to square it. And that could be your final answer. The only thing that I want to go ahead and do is get rid of this negative exponent because you don’t usually like to have negative exponents in your answer. So I’m going to go ahead and say that the derivative is -2/x2 over and what I did, remember before we moved the x from the denominator to the numerator we flip the sign on the exponent. This time we’re going to move it from the numerator to the denominator of the top part here and flip the sign from a negative to a positive. So we can move it in either direction as long as we flip this from a positive to negative, just change it. So we have that and then we have the bottom the same 1 – 2/x2 and that’s our final answer.
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