Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

Question

ƒ(x) = (x)/(x+1)

Accepted Answer

Welcome back, everyone. For the function F of X equals x divided by x + 7, calculate the factored form of the difference quotient F of X plus H minus F of X divided by H. So first of all, if we look at the difference quotient, we want to identify F at a point X plus H. Now what we have to do is simply substitute X plus H for every X that we see. So we have one in the numerator, it becomes x plus H, and now our denominator becomes X + H+7. Our F of X is the original function, which is X divided by X + 7. What we want to do is subtract these in the numerators. We're going to write F of X plus h minus F of X divided by h to indicate what we are calculating, and then we're going to substitute the first term which becomes x plus H. Divided by x plus H + 7, we are subtracting x divided by x + 7, and then this difference is going to be divided by H. So we have all of the terms that we need. We now want to simplify and the way we do this is simply by finding the common denominator of the two fractions in the numerator. So we're going to take X plus H. Multiply it by the denominator of the adjacent fraction, and that's x + 7, right? So we are multiplying by x + 7 and then we're subtracting X, which is going to be multiplied by the other denominator X plus H + 7. All of this is going to be divided by the common denominator, which is. X plus H + 7. Which is multiplied by X + 7, right? So the product of the two denominators, and now because we have a double division, then everything is divided by H, we can just write a single denominator which with H in front of the two factors. Now what we're going to do from here is apply the foal property, so we have X plus H multiplied by x + 7. So X multiplied by X gives us X squared, x multiplied by 7 gives us 7 X. so we're adding 7 X, then plus H X. And then plus 7 H. Well done. And now we are going to distribute the terms for negative X multiplied by X, right? This gives us negative X squared, then we're multiplying negative X by H, which gives us negative HX, and then negative X multiplied by 7. This gives us -7 X. We are going to rewrite the denominator H multiplied by x + H+ 7 multiplied by x + 7, and now let's carefully analyze the numerator. So x2 minus X2, this gives us 0.7 X minus 7X this gives us 0, and then that's it. What we can notice is that we can factor out the H and this leaves us with X. Plus 7 because we have HX plus 7H and then we are subtracting HX, right? So this means that we are also subtracting X. So we could have canceled out HX as well. We can show that. And we end up with 7 H. In the numerator. So we're going to write 7 H and then our denominator has H in front, X plus H + 7 multiplied by X plus 7. Now we can cancel out the H, and this gives us the final answer in its simplified form, which is going to be 7 divided by X + H+ 7. This is our first factor, and the second one is X + 7. This is our final answer, so we're going to label it and thank you for watching.

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = (x)/(x+1)

Key Concepts

Difference Quotient

Function Evaluation

Simplification Techniques

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