in this video we're going to cover some of the standard derivative formulas that we're going to be using throughout calculus and throughout your engineering career note that these formulas are standard in the grade 12 ontario curriculum as well as most curriculums across canada so we are going to move through this content fairly aggressively if you find yourself in the position where this is new for you i would encourage you as quickly as possible to get into the practice of this and you can use that either through textbook readings or through the qn prep site that was available over the summer as well those are both excellent resources to go a little more in depth on the concepts that we're doing here and the skills that we're developing here many of you will have seen derivatives already introduced as slopes and you can see that hiding in the definition formulas for the derivative these are both the root of the idea of the derivative in a graphical sense where we have a change in a height divided by a change in a width on a graph and then we ask the question what happens to that ratio what happens to that slope as we make the step the x step as small as possible however going through this process of a complicated numerator denominator and then taking a limit gets a little bit tedious after a while and when we're working with very common functions like polynomials or trig functions or exponentials maybe nested with other functions inside it turns out that this practice of calculating limits and doing ratios turns out with certain patterns or returns a certain patterns and so what we end up doing instead is bottling that calculation up into a more standard notation for commonly used functions if we have to we have the definition but day-to-day practice is simply to apply these derivative rules pattern rules to the functions that we use all the time so what we're going to do now is simply explore that set of rules and build up that skill set of working with functions and calculating the formula for their slopes here is your first review of the derivative rules the derivative of a constant is zero so for example if we have the derivative of five if you think of five as a function it would just be a horizontal line what's the slope of that line it's zero polynomials are built out of powers of x to some power this rule here is paraphrased at least i paraphrase it as bring the power down front and then subtract one from the power if you do that if you say that mnemonic i find that more intuitive or easier to remember than this more formulaic approach for example if we take the derivative or ask what this slope formula is for x cubed we bring the power the 3 down front and we'd get x to the three minus one one lower power x squared this defines the slopes of the graph of x cubed the next two rules are ones that most people just apply sometimes that recognizing that it might be an issue but if we have two things added together you can take the derivative of each one of them separately that's totally fine same with subtraction the one that tends to cause grief is if we have a constant but not by itself the key thing is here function by itself here we have a constant that's multiplying another function so something that is changing and there the rule is we can bring the constant out front and keep it as a multiplier keep the constant as a multiplier this becomes much clearer when we go through some exercises and some practice for example we have two things added together so we can find the derivative of one first then add the derivative of the second term the derivative x the four bring the power down front and take one lower power so we get x cubed plus the derivative of 3x squared this 3 is a multiplier of another function so it stays around so we get 3 and then times the derivative of x squared bring the 2 down keep the x to the power 1 and we'd usually tidy that a little bit to simply 6x to the power 1. we don't need to show the power this derivative here looks a little hairier fortunately the first thing we can do is rewrite it in a form that makes those rules easier to apply and by that i mean right roots and in general fractions as powers so remembering that d dx is the verb it says take the derivative of all of this we're not going to take the derivative yet so we still have the take the derivative of and then we say x square rooted is the same as x to the one-half the pi is fine that's already as a power and now we can actually take the derivative because we have written it in a way that is easier to apply the rules from the last page this 2.6 is a multiplier this pi is a multiplier and we'll get to the 4 in a second two point six x the one half the two point six stays x to the one half we bring the one half down front and then we subtract one from that power half minus one is negative a half and that finishes that first term because it's then subtracting something else so minus pi is a multiplier of x cubed so the pi stays has a multiplier and we bring the 3 down front we subtract 1 from the power we get 3x squared last but not least though we have constant by itself 4 as a function has 0 slope so when we take the derivative which is calculating the slopes we get a zero just be aware of that distinction multipliers of functions they stay around multipliers of one or constants their derivative is always going to be zero and then again we can tidy this up we're not too worried in the course about simplifying derivatives because it depends on what you're doing next but some basic tidying of constants like this is just good practice whether you want to leave this x to the negative one half or just recognize that that could be written as 1 over root x that's good practice with your handling exponents doesn't really matter and there's not much we can do with the 3 pi x squared except usually the numbers get written before the pi symbol now consider the derivative of negative 3x squared minus 1 over x squared take a moment to think about these options or do the calculation yourself and then we'll take a look at it all right if we do our derivative calculations we'll do this the long way except the one thing we're going to do is rewrite this as a single power this is x to the negative 2. so if we're careful with our rules the negative 3 is a multiplier the derivative of x squared is bring the power down front 2x to the power 1 minus bring the power down front applies again here subtract one from negative two makes it more negative and we get negative three and then when we tidy up we're going to have negative six x plus two over x cubed we have the negative exponent so we can write that x cubed in the denominator then we can scan down our list and go we need the negative 6x that's great plus 2x cubed that makes b the correct answer here the next most commonly used functions after powers of x are the e to the x function and log function and also other bases so 2 to the x 10 to the x log base 2 log base 10 commonly used as well for calculus you can see why e to the x is special when we take the derivative of the function e to the x and this is a bit surprising if we graph it here's e to the x it turns out that the way this function is built and the fact that e is 2.71 et cetera et cetera et cetera that carefully crafted number guarantees that the slope at any point also is given by exactly the same formula that's a bit surprising that's a bit weird or it should feel weird because values and slopes generally are not this closely related in fact the exponential function is special in that it has this relationship the derivative of the values or the slope at any point is given by the same function that you started with you can see even for other bases that that's not the case it's almost the same thing if we have the derivative a to the x we get the same a to the x again but then we get an extra scaling factor of the lawn of the base now that actually applies here as well if you want to you could write times the lawn of e and use the same rule but the lawn of e is just one so we can't see it when we do the e to the x derivative this is the general rule applies all the time for any base the e to the x one just happens to be the simpler case because that extra month that extra scaling factor disappears switching gears if we have the lawn the natural logarithm its derivative is 1 over x not something you see coming proving that's actually a bit of a challenge but it's a very simple rule to memorize same idea with the log of some other base so if we had log base a we still get the 1 to the x so same pattern as the exponentials we get the same root function same basic function involved but then we have the scaling factor this time here in the reciprocal or in the denominator the scaling factor 1 over log of a so recapping derivative e to the x is e to the x through the log of x one over x those are the main two that we use but there's they're related cousins with a new base and all they are are the same rules except for the long a multiplier for exponents and a lot a divider for logarithms let's practice with that again here we see some multipliers the 10 and the 4 here and then we have the exponential function and a power so we're trying to distinguish those two let's take the derivative of the first term this is a form of a to the x the base is constant the exponent is changing with x and so we get the same exponent back times the lawn of the base that's our derivative rule for a to the x rules then this one here x is in the base and what's constant is the power p here's the power of four so we use the other rule so the 10 multiplier that stays then it's bring the power down front subtract one from the power and there's not a lot we can do with this either but we might just tidy it up a hair for that second term taking the 4 and the 10 and multiplying out to be 40. this example highlights how careful you have to be about where the variable is it has a huge impact on what rules you use and only one rule will ever apply at a given time similarly here we have e to the x its derivative is just e to the x if you wanted to you could include the law of the base but we know that's one so we'll just erase that it has no effect then we have a logarithm the derivative of every logarithm is one over x if it's log base 10 we have that scaling factor in the denominator of one over lon 10. now there's not a lot more we can do with the exponential and log rules until we get to the more advanced combination rules like product quotient and chain rules which we will get to and explore these a little further