Find f(x + h) − f(x)/h and simplify. f(x) = x⁴+7

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Hey everyone here we are asked to use the difference quotient formula F of X plus H minus F of X. All divided by H. To find the difference quotient of the function F of X is equal to X to the fifth power minus two. Now here to solve this problem, we first need to look at our formula here in, we need to notice that we already have this F of X function but we need to find this F of X plus H function. So doing this F of X plus H. All we have to do is substitute in X plus H four X from our given function. So doing this, we have the quantity of X plus H to the fifth power minus two. And with this we need to simplify this function here. So that means we need to take this quantity of X plus H to the fifth power and expand it. So to expand it, we need to recall the binomial theorem which gives us a formula to expand binomial and tells us that for any positive integer N. That the quantity of a plus B to the end power is equal to n 10 0 times A. To the power of n plus n one times A. To the power of n minus one times B plus n. Two times A. To the power of n minus two times B squared plus n. Three times A. To the power of n minus three times B. Cute. And we continue to expand the this expression until we reach the term n n times B to the power of N. And now before we can use this formula, we need to identify these variables here in relation to our expression. So remembering that our expression that we're expanding is the quantity of X plus H to the fifth power. So now identifying the variables we can start with. Well A is just the left hand side term from our expression. So A will be X. Which leaves the other term to be B. So B is going to be a church. And lastly our end value is this power to which our binomial is raised to. So our end is going to be fine. And now with this information identified, we can go ahead and start expanding. So again again we're expanding the expression where we have the quantity of X plus H to the fifth power. Beginning with our first term. Starting with the coefficient we need to remember that N is five. So we have the value five and our second value is zero for our first term and then we multiply by A which is X. And raise it to the end power which in this case is five. Now we can move on to our next term where we have 51 for the coefficient times X. To the power of five minus one times B. Which in this case is H. Moving on to the next term we have 52 times X to the power of five minus two times H squared plus Our next term +53 times x to the power of five minus three times H cubed plus our next term we have +54 times X to the power of five minus four times H to the fourth power. And then plus our next turn we have the coefficient + and we can see that this will be our last term because looking at our formula, we have our coefficient as an N. And in this case it is +55. So we are left with +55 times H to the fifth power. And so now we have fully expanded our expression. So we just need to start simplifying and we need to recall that to simplify these coefficients here the coefficient and R is equal to And factorial divided by the quantity of n minus r factorial times R factorial. So using this formula we can go ahead and begin simplifying by first writing out our coefficients. So beginning with our first term, our coefficient turns into five factorial divided by the quantity of 5-0 factorial times zero factorial and this is the coefficient of X to the fifth power plus our next term we have five factorial divided by the quantity of five minus one factorial times one factorial. And then our next two terms our next two variables here in this term are X to the power of five minus one times age plus our next term. We have five factorial divided by the quantity of five minus two factorial times two factorial and this is the coefficient of X to the power of five minus two times H squared. Now four. For our next term we have five factorial divided by the quantity of five minus three factorial times three factorial and this is the coefficient of X to the power of five minus three times H cubed plus our next term we have five factorial divided by the quantity of five minus four factorial times four factorial. And this is the coefficient of X to the power of five minus four times H to the four power. And for our last term we have the coefficient of five factorial divided by the quantity of five minus five factorial times five factorial and this is the coefficient of H to the fifth power. So now we just need to continue to start simplifying here. So for our first term we can perform the subtraction in the denominator of the coefficient. So we have five factorial, define it by five minus zero is just five factorial times zero factorial and then our X term is already simplified. So this coefficient is multiplied by x to the fifth power. Moving on to our second term. We have five factorial divided by five minus one gives us four factorial times one factorial. And our x term in this second term here can be simplified to X to the fourth power and then we have times H moving on to our third term we have five factorial divided by five minus two, gives us three factorial times two factorial. Then our X. Here the power simplifies to three. So X is cute times H squared plus our next term we have five factorial divided by two factorial times three factorial and this is the coefficient of X squared times H Q plus our next term simplifies to five factorial divided by one factorial times four factorial And this is the coefficient of X times H to the fourth power plus our last term we have five factorial divided by zero factorial times five factorial and this is multiplied by H to the fifth power. So from here we just have to keep simplifying our coefficients. Beginning with our first term, we see that the five factorial cancel. And we need to recall that zero factorial is simply equal to one. So our coefficient for the first term is just one. So we have one times next to the fifth power. Moving on to our second term, we see that we have four factorial in the denominator. So we can go ahead and expand the numerator until we also have four factorial. So with this, our numerator becomes five times four factorial and this is all divided by four factorial times one factorial. And this is the coif efficient of X to the fourth power times age moving on to our third term. We have a three factorial in the denominator. So again we can expand our numerator until we also have a three factorial. So doing this, we have five times four times three factorial. All divided by three factorial times two factorial. And this is the coefficient of X cubed times H squared plus our next term we have the same coefficient as our previous term. Again pointing out this three factorial in the denominator allowing us to expand this numerator. So we have five times four times three factorial, all divided by two factorial times three factorial and this is the coefficient of X squared times H cubed plus our next term. Again we have a four factorial in the denominator. So our numerator can be expanded to five times four factorial all divided by one factorial times four factorial. And this is the coif of X times H to the fourth power. And now with our last term here it is just like our first term where the five factorial is canceled. And we are left with the coefficient of one. So we have one times H to the fifth power. And now we can continue to simplify our coefficients here. Looking at our first term, we can just drop the one coefficient since it's already implied. So our first term is just X to the fifth power plus our next term. We see that the four factorial is cancel. So we're left with five divided by one factorial which is just five times X. To the fourth Power times age plus our next term we see that the three factorial is cancel. So we're left with 20 divided by two factorial which is just 20 divided by two giving us the coefficient times X cubed times H squared plus our next term we have the same coefficient of times X squared times H cube moving on to our next term. We have the four factorial is cancel out. And we are left with the term five times X times H to the fourth power. And again with our last term it's the same as our first term. We can just drop the one coefficient and we are left with H to the fifth power. Now with this expression expanded we can rewrite our F of X plus H function. So recalling that we had F. Of X plus H. As the quantity of X plus H to the fifth power minus two. Now incorporating the expansion we simplified. We have F. Of X plus H. Is X to the fifth power plus five X to the fourth Power times eight plus 10 X cubed times H squared plus 10 X squared times H cubed plus five X times H. To the fourth power plus H to the fifth power minus two. And so with this we have our two functions for our difference quotient formula. So all that is left to do is input them into our formula. So again recalling that our F of X function is just X to the fifth power minus two. So again using both of these functions into our formula, starting our numerator with our F of X plus H function. We have X to the fifth power plus five X to the fourth power times H plus 10 X cubed times H squared plus 10 X squared times H cubed plus five X times H. To the fourth power plus H. To the fifth power minus two. And now we have minus the F of X function. So we have minus the quantity of X to the fifth power minus two. And again this entire expression here these two functions are divided by H. So now we just have to start simplifying again. So we can begin by just distributing this negative sign. So doing this we have X to the fifth power plus five X to the fourth power times H plus 10 X cubed times H squared plus 10 X squared times H cubed plus five X times H. To the fourth power plus H. To the fifth power minus two minus X. To the fifth power plus two. And once again this entire expression here is divided by H. And from here we can start simplifying to by seeing which terms cancel. So here we can see we have a positive two and a negative two. So they will both cancel. And lastly we have a positive X To the fifth power and a negative X to the fifth power. So they will both cancel as well. So now we have to rewrite our expression and divide each term by H. To get our simplified solution here where we have five X to the fourth power plus X cubed times H plus 10 X squared times H squared plus five X times H cubed plus H. Two. The fourth power. And so with this we have finished finding our difference quotient and found our simplified answer. And we are left with answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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