let's talk for a little bit about elasticity of demand basically elasticity refers to how sensitive a product is to a change in price so in other words if you increase if a company increases the price of a product are people still going to buy it or are people going to be turned off and say no if you raise the price i'm not going to buy it anymore so that's the the general idea behind elasticity of demand and the equation that we use is this one here at the top of the screen we calculate elasticity as a negative x where x represents the price multiplied by the derivative of the demand function typically the demand function is given just like it is here divided by the original demand function so in order to calculate elasticity we have to have the derivative of demand and the original demand function here we have the original demand function so this is going to end up going in the denominator of elasticity for the numerator we need to get the derivative of this demand function so getting the derivative is going to involve the quotient rule so we have the second times the derivative of the first minus the first times the derivative of the second over the second or the denominator squared all right let's just keep our denominator as is in factored form distributing this 4 and then distributing this 20 we're going to have 80 x plus 28 and then watch out with that minus in the middle minus 80 x and then minus 6 000. simplifying across the top and keeping the denominator the same we then have the 80 x's wipe each other out plus 28 and minus 6000 is going to turn into 59 72 negative so this is our simplified derivative of the demand function that's our derivative on the demand function that whole thing goes into the numerator of the elasticity function and remember the denominator of the elasticity was the original function so when i put that together big picture this big fraction is going to look like i have negative x from the elasticity function and then times and here is where i need the derivative of demand which was this one so that's going to go in here and the whole thing is going to sit on top of the original demand function this one right here going into the big picture denominator so here's my derivative of the demand function and here's the original demand function so to simplify this i see that i have a negative x times negative 5972 so that's going to become a positive 5972x and then remember when you divide by a fraction you can flip and multiply so since i'm dividing right here by this big fraction i can flip and multiply so now i'm going to multiply by 20 x plus 7 over 4x plus 300 and the parentheses are never going to hurt and they often help all right since i'm multiplying i can cancel the quantity 20x plus seven one for one so i'll take away one of these and then i'm left with for my elasticity equation i have 5972 x in the numerator over 20 x plus 7 quantity one of those left over in the denominator now if i take a look at this right here technically your parentheses should never have terms that share a common factor so 4x and 300 have a common factor of 4. so i can pull that 4 out factor out that 4 and if i factor out that 4 i'm left with x plus 75. that's just proper procedure when you have factors you want to make sure that they don't have a common factor if they do you're supposed to pull it out in so doing since it's all multiplication i now see that i can cancel top and bottom i can cancel a 4. so if i divide this by 4 that's 1 if i divide this by 4 i'm going to have 14 93 x in the numerator over divided by the quantity 20x plus 7. 20x plus 7 is multiplied by x plus 75 in the denominator so this right here is my simplified elasticity formula that will be the answer to question a for b i then just want to use that elasticity formula when x is 4. so simply put a 4 in everywhere there's an x and work that through so if i do the elasticity of x equal 4 putting a 4 in here a 4 in here and a 4 in here and work that through i should get 1493 times 4 is 5972 and then i put a 4 in here i get 87. if i put a 4 in here i get 79 87 multiplied by 79 is 68 73 and if you put this in your calculator you can confirm that it's already in lowest terms believe it or not and the way just to remind you how to do that on your calculator you put 5972 divided by 68 73 and hit enter and you'll get a decimal and then 59.72 divided by 68.73 you get a decimal of about 0.8689 if you hit the math it's on the left hand side of the calculator hit math enter enter and you'll get back 59.72 over 68.73 that's how you know it's in lowest terms but if you observe the numerator is smaller than the denominator so that means this fraction is smaller than one and if elasticity is less than one we call that in elastic let's take a look at the properties for inelastic inelastic means that raising the price at least raising it a small amount is probably not going to deter that many people from buying it whatever the product is the people want it and changing the price a little they're going to pay they still want the product so basically if it's inelastic you're going to get more revenue because raising the price is going to bring in more revenue and most people are going to still keep buying it so total revenue would be increasing