hi so for today we're going to talk about rules for finding the derivative so we have rules for finding the derivative so for our number one rule is the derivative of constant so always keep in mind that the derivative of constant is always equal to zero so it means if you have a Y that is equal to three if you get the derivative of that dy over DX the derivative of Y with respect to X it's always equal to zero so here is the formula D over DX of any constant C is actually equal to zero okay so if you're asked to find the derivative for example let's say again Y is equal to 1,000 the derivative of Y with respect to X is always equal to zero okay so for our number two is what we call the power rule the power rule has this formula dy D over DX of X raised to any number n then it is absolutely equal to n times X raised to n minus one okay that is we're going to get the power then multiply it to our original X but we're going to subtract one from its original exponent so let's say for example we have Y is equal to X cube okay if we're going to get the derivative of Y with respect to X we get the exponent three then multiply it by the function X cube but we have to apply minus one because the formula says M times X raised to n minus one so that we have this three X raise to 3 minus 1 is actually three x squared and we now get the derivative of this function y is equals to X cube so another example is for example we have a function that has X raised to the fifth okay so we have dy over DX is equal to 5 times X raised to 5 minus 1 or in other words simplifying this 5x times X raised to the fourth okay so very simple formula they the power rule parameter is the derivative of a constant times a function times a function so it says here the formula dy over DX is actually equal to D over DX of the function constant times or certain f of X so we have here since the derivative of cost time is we have to since this is constant C we can actually isolate that and get the derivative of f of X ok so for example we have here an example so for example find the derivative of Y is equal to 3 times X raised to 6 so this is our constant three so all we have to do to find the derivative of that is to apply that formula so what are we going to do is three times dy over DX of X raise to 6 right so in order to get that we have three apply our formula six times X raised to 6 oops X raise to 6 minus one so we have applied the power formula so to simplify we have 3 times 6 that is 18 X raise to 6 minus 1 is 18 times X raised to 5 so if we have a constant simply we're going to get the derivative of this function and isolate this constant okay so another example is this for example we have here y is equal to 2x squared okay in order to get the derivative of this we have to leave the constant itself and get the derivative of the function x squared so we have here to find the derivative of x squared by power formula we get the exponent we multiply it then the original function X raised to minus 1 that will be equal to 2 times 2 that is 4 times X 2 minus 1 is X raised to 1 or in other words X so that we have the answer for X okay so how about the fourth rule the derivative of sum and difference so in this case we shall be applying the formula dy over DX is equals to D over DX times u plus or minus V and that is equal to the D u over DX plus or minus DV over DX rain u and V are two turns okay so for example we have here this y is equal to 3x squared plus 2x minus 1 and find the dy over DX of that function so in order to get the dy over DX or the derivative of this function y we have to we have to evaluate the individual derivatives of this function so we have the first turn that is the derivative of a constant 3 times a function x squared and also for the second term that is the derivative of a function times a constant function X and a constant 2 and in this third term that is simply the derivative of a constant so if you wish to get the derivative of that so we have 3 times we're going to get T over DX of x squared plus 2 the derivative of the function X minus the derivative of the constant 1 so we're going to get that 3 and by power rule this is time spoon times X raised to 2 minus 1 and plus - that is times 1 because the exponent of this X is 1 so and times X raised to 1 oops 1 minus 1 minus the derivative of constant is again 0 so we have here 3 times 2 that is 6 X raised to minus 1 that is simply X plus 2 times 1 this this X raise to 1 minus 1 is X raise to 0 that is 1 so simply 2 times 1 times 1 that is 2 so we have now the derivative of this function so another one how about if we have this so we have Y is equal to turns of x cubed minus 4x plus 86 so same same formula same approach we have to get dy over DX this is a constant and a function this is a constant again in a function so we have to get 2/3 the derivative of x cubed minus 4 times the derivative of this function X plus D over DX of this intersects constant so 2/3 bring down 3 because that is the exponent multiply it the X cubed minus 1 so our rule again so minus 4 the derivative of this X is simply 1 right we're going to evaluate it yeah okay so plus the derivative of this constant again the derivative of constant is yes you are right that is 0 so as you can see here 2/3 times 3 simply the 3 will cancel and then we will be having puh times X raised to 3 minus 1 that is x squared minus 4 and that should be our answer ok so so the fifth rule is the derivative of a product so it says here the power formula is dy over DX we wish to find the derivative of prada d over DX of U times V where u and V are both function so we have to apply this formula u is ik times the DV over TX plus the V D u over the X ok so we have this U and V so first we have to get or to copy first the function U then let's start the derivative X that will be our first term plus copy V then we're going to get the derivative of U with respect to X so one example is this V is X plus 1 times X plus 2 all right we can use foil in order to get this to perform this product but using this derivative of a product we can simply use the formula let's say dy over DX so what are we going to is this will be our U and this will be our P so it sets in the formula we need to copy the first one X plus 1 then we're going to get derivative get the derivative of the second one x plus 2 plus copy the V this is our V X plus 2 then we're going to get the derivative of the you which is absolutely X plus one so in other words copy the first this first and then get the derivative of this multiply it that's your first term okay plus copy the second term and then what are we going to do is multiply it by the derivative of the first so in this one we have X plus 1 and the derivative of this is actually what the derivative of X is 1 and the derivative of constant is actually 0 so plus X plus 2 the derivative of X is actually 1 and the derivative of 1 is actually oops I'm sorry I need to erase this okay then the derivative of one or this one is absolutely zero so that is a constant so again we get the derivative of X which is equal to 1 we have a derivative of 1 which is equal to 0 so simplifying we have X plus 1 simply this will be multiplied to 1 because 0 nothing to do with that so X plus 1 plus simply X plus 2 is multiplied to 1 so we will be having X plus 2 here so to simplify that we have X plus 1 plus X plus so we have the answer combine like terms X plus X that is 2x plus 1 plus 2 that is equal to 3 so another way to get this is what I am Telling You a while ago is to get the product of this using the foil method okay so if we're going to get the derivative of that we simplify that first so into a single function we multiply this 2 so what are we going to have is the oil apply the foil method first x times X that is equal to x squared outer x times 2 that is plus 2x and our inner is 1 times X that is equal to X and last one times 2 that is plus 2 then we're going to get the derivative of this so if you get derivative of this dy over DX so we apply the power formula here so 2x plus 2k plus 1 Y power formula for here x squared - X then this is the derivative of a constant times a function X derivative of constant we're going to get it out of the derivative then derivative the function X will result to 2 and the derivative of this X is 1 plus the derivative of this 2 is actually a constant that is equal to 0 to simplify we have not the answer simply 2x plus 2 plus 1 is not equal to 2x plus 3 which is the same the same as this ok but sometimes there will be given function in terms of Y that is really hard to multiply or really I would say I'm consuming so you would really want to use this protocol then rather than getting fresh the product then apply this formula of power rule and the derivative of a constant times a function so let's try another formula we have Y is equal to X cubed plus 2x times 2x minus 1 so you're going to get a derivative of this y prime that is equivalent also to DUI over DX simply this is our u this is our V so simply copy the you first or the first term X cubed plus 2 plus 2x times the D over DX of 2x minus 1 that is the derivative of the second plus copy the second which is 2x minus 1 then we're going to get the derivative of U or the first okay so we have X u plus 2x so here we go so we have X cubed plus 2x times the derivative of 2x is actually what it is equal to 2 okay and the derivative of this negative 1 or minus 1 since that is a constant that is minus zero plus 2x minus 1 times the derivative of this so we can apply power formula in this one so we'll be having 3 x squared plus 2 okay so the derivative of this is absolutely 2 because this is a function X multiplied by constant the derivative of this is 1 2 times 1 that is 3 by 2 and this is a product power rule I mean okay so simplifying our answer we have X cubed plus 2x times 2 plus 2x minus 1 times 3x squared plus 2 okay so in order for us to find that so we're going to simplify 2x cubed plus 2 times 2x that is 4 X plus we're going to find the foil of this thing so first 2x times 3x squared that is 6x you outer 2 X times 2 that is equal to 4x inner negative 1 times 3x squared that is negative 3x squared and finally last term negative 1 times 2 that is equivalent to minus so combining like terms so we have here 2x cubed plus 6x cubed that is simply 8x cubed so we have a second degree here so we can write it here so another one 4x plus 4x again we're combining like terms here 4x plus 4x that is 8x and then finally the constant term is minus 2 so our answer should be 8x cubed minus 3x squared plus 8x minus 2 like this shall be our answer so another one for product rule we have here the given X cubed minus 3x squared multiplied by x squared plus 4x plus 2 so same concept we will apply the product rule here so again copy first the first that is that shall be our U and this shall be our V first turn minus 3 x squared times we're going to get a derivative with respect to X of the V which is x squared plus 4x plus 2 plus copy the V or the second term plus 4x plus 2 then we're going to get the derivative of x cubed minus 3x squared that is the derivative of the first so this case we should copy only here then we're going to get the derivative of this this is a combination of a Prada or our and this is a derivative of constant times a function and this is the derivative of a constant so the derivative of x squared is simply by power rule 2 X and the derivative of 4x is simply 4/4 and the derivative of this constant 2 is actually equal to 0 so plus x squared plus 4x plus 2 times the derivative of this since this is a power rule so 3x square yes you got it right so - this is a combination of power would add a constant so we have 3 times 2 because of power rule 6 X raised to 2 minus 1 so we will be having 6x okay so simplify X cubed minus 3x squared we have 2x + 4 and then for this we have x squared plus 4x plus 2 times 3x squared t XY minus 6x so in order to simplify we need to apply foil first X cubed times 2 X that is equivalent to 2 X raise to 4 outer X cubed times 4 that is plus 4x cubed inner negative 3x squared times 2x that is equal to negative 6 X cube and last negative 2x squared times 4 that is minus 12x squared so it gets the computation gets long okay so plus that is actually foil or not foil but simply the normal computation for product of this to function so we have 3x squared times x squared that is 3 X raise to 4 and 3x squared times 4x that is plus 12x cubed so 3x squared times 2 that is plus 6x squared ok then negative 6x times x squared that is equivalent to that is equivalent to negative x raised to 3 so negative 6x times 4x that is negative 24x squared and that's the negative 6x times 2 that is negative 12x so to simplify we're going to combine like terms ok so we have first the highest exponent we can get it here so this is the highest exponent so 3 X raise to 4 so next is we have rest of war that is oh my mistake so that should be now we're wrong this should be we can simplify to X raise to 4 then plus the X raise to 4 that is equivalent to 5 X raise to 4 then we have here cubed plus 4x cubed minus 6x cubed that is negative 2x cubed plus 12x cubed that is 10 X cubed minus 6 X cube that is plus 4x cubed ok and for our x squared negative 12x squared plus 6x squared that is equivalent to negative 6x squared minus 24x squared negative 6x squared minus 24x read that leads to negative 30x squared and finally for our X so we should copy because we don't have anything to simplify with X so this should be our final answer okay so let's go now to our rule so our last rule for this the joy is actually the derivative of a motion so this will be our formula for that dy over DX is equal to the derivative with respect to F of a function that is in goshen form is equals to V times D U over DX minus u times DV over DX all over by V squared okay so this will be our formula when we have a the numerator you and denominator of T so what are we going to do first is that we're going to get first the the denominator then multiply it to the derivative of the numerator - the numerator multiplied by the de the derivative of the denominator DT over D X all over we're going to square the denominator or in other words that is we have a mnemonic for that low derivative of high minus high derivative of low / low which is the denominator squared so let's try some problem so for example we have X Y is equals to X over X plus 1 so it is obvious that our you here is the X and our B here is the denominator which is X plus 1 so in order to get a derivative of our function so we're going to apply this formula or simply this so what is that lo the denominator times the derivative of the hi which is actually the numerator minus copy the hi which is the numerator times derivative both low which is X plus 1 divided by Rho squared which is X plus 1 squared so let's recap so we have here lo which is the denominator lo times the derivative of hi - hi which is the numerator times the derivative of no so divided by the low squared or the B squared the denominator or base right so no derivative non high high derivative off low divided by X plus one squared so we shall have the answer X plus 1 times the derivative of X which is simply 1 minus x times the derivative of X is 1 plus the derivative of 1 is 0 divided by X plus 1 squared so we have the answer X plus 1 times 1 which is simply X plus 1 minus x times 1 which is X divided by X plus 1 squared to simplify we X will cancel X - X so we are left with 1 over X plus 1 squared so this will be our answer hey so another problem derivative of a potion we have 2x x squared plus 1 so again we will be applying the low D high I suggest you memorize this low D high minus high D low / no squared okay so first we need to get the low which is the denominator x squared plus 1 multiply it by the derivative of the high D over DX of 2x minus high which is 2x times the derivative of low D over DX x squared plus 1 divided by Rho squared x squared plus 1 raised to so we have here x squared plus 1 times the derivative of 2x is simply 2 - 2 X the derivative of this we have we have minato evaluate first this x squared is 2x plus the derivative of the constant is zero so / still x squared plus 1 squared so we have now x squared + 1 times 2 that is 2x squared plus 2 minus 2x times 2x that is 4x squared divided by x squared plus 1 squared so we have now the answer - x squared minus 4x squared that is negative 2x squared + 2 / x squared plus one squared okay so we can simplify this further by factoring out poop so we have 1 minus x squared divided by x squared plus 1 squared so this shall be our answer if you find this video helpful don't forget to click like and subscribe to my channel thank you so much for listening and God bless