 So in a previous video we've learned that we can use the quotient rule to calculate the derivative of a function which has quotients or divisions or ratios of some kind kind of like the example you see here on the screen in this video I do want to point out that it's it's important to remember that you don't always have to use the quotient rule That is to say when a function has a quotient structure by means you can use the quotient rule But sometimes the use of algebraic or trigonometric identities can simplify The function in such a way that the quotient could be removed and perhaps you can compute the derivative in a simpler way So consider the example you see on the screen here f of x equals 3x squared plus 2 times the square root of x all over x Well since the denot I mean it is a it is a fraction right that you're dividing by x right here The quotient rule will be appropriate here You could do low d high minus high d low in all that business, but recognizing that the denominator is a monomial We actually can distribute a monomial denominator And so this is the same thing as 3x squared over x plus two times the square root of x over x For which with the first term you can see that since you have an x on top and bottom you can cancel And so the first term would actually become Just a 3x and then the second term Oh, okay I recognize that if I think of the square root of x not as the square root of x But instead as x to the one half power that is I think of as a power function Then when you subtract the powers you're gonna have negative one half minus the first power right here And so you get two times x to the negative one half power And so the function f itself is just a linear combination of power functions taking its derivative It's essentially the same thing as taking the derivative of a polynomial So just using the linearity of the derivative and the power rule We take the derivative of 3x and we just get a 3 derivative of x of course is 1 and then for the next one We're gonna get two times negative one half x to the negative three halves Likewise by the power rule two times one half gives you one And so the derivative here turned out just to be 3 minus x to the negative three halves Or if you prefer you can write this as 3 minus 1 over You know x squared of x or something all of those things are equivalent to each other I'm not gonna word too much about that. We'll just leave the original form right there And so we can see that this function is much easier to calculate the derivative if we can simplify The rational expression beforehand and so the takeaway I want you to skip from this example is not just that sometimes we can avoid the quotient rule is that in general The proper use of algebraic identities and trigonometric identities can simplify functions to make the calculus calculations much much easier This is the whole point of knowing these identities, you know many students like in a trigonometric class They're like trigonometric identities are the worst thing ever know what they are tools They're tools to simplify trigonometric expressions much in the same way that algebraic identities like factorization formulas and the like Help us with algebraic settings It's not that a trigonometric identity or algebraic identity is poison or a curse upon you It's a tool and so the more tools you have the more projects you can work on and I get it When you have a lot of tools and you don't know what all of them do it gets overwhelming But be aware like in this example a good Algebraic simplification can make the calculus much much easier if you're not convinced try computing the derivative of this thing Straight using the quotient rule and compare what we did in this example and what you did yourself