Worked example: Quotient rule with table

let F be a function such that F of negative 1 is equal to 3 f prime of negative 1 is equal to 5 let G be the function G of X is equal to 2x to the third power let capital F be a function defined as so capital F is defined as lower case f of X divided by lower case G of X and they want us to evaluate the derivative of capital F at x equals negative 1 so the way that we can do that is let's just take the derivative of capital f and then evaluate it as at x equals 1 and the way they've set up capital F this function definition we can see that it is the quotient of two functions so if we want to take its derivative you might say well maybe the quotient rule is important here and I'll always give you my aside the quotient rule I'm going to state it right now and it could be useful to know it but in case you ever forget it you can derive it pretty quickly from the product rule and if you know it the chain rule combined you can get the quotient rule pretty quick but let me just state the quotient rule right now so if you have some function defined as some function in the numerator divided by some function in the denominator we can say its derivative and this is really just a restatement of the quotient rule its derivative is going to be the derivative of the function in the numerator so d DX f of x times the function in the denominator so times G of X minus minus the function in the numerator minus f of X not taking its derivative times the derivative of the function in the denominator D DX G of X all of that over so all of this is going to be over the function in the denominator squared so this G of x squared G of X G of x squared and you could use multi different types of notation here you could say you could say instead of writing this with a derivative operator you could say this is the same thing as G prime of X and likewise you could say well that is the same thing as f prime of X and so now we just want to evaluate this thing and you might say how do I evaluate this thing well let's just try it let's just say well let's we want to evaluate F Prime when X is equal to negative 1 so we can write F prime of negative 1 is equal to well everywhere we see an X let's put a negative 1 here it's going to be F prime of negative 1 F prime lowercase F prime is a little confusing lowercase F prime of negative 1 times G of negative 1 G of negative 1 minus F of negative 1 F of negative 1 times G prime of negative 1 G prime of negative 1 all of that over all we doing that same color so I take my color seriously all right all of that over G of negative 1 squared G of negative 1 squared now can we figure out what F prime of negative 1 F of negative 1 G of negative 1 and G prime of negative 1 what they are well some of them they tell us outright they tell us F and F Prime at negative 1 and for G we can actually solve for those so let's see if this is well let's just first evaluate G of negative 1 G of negative 1 is going to be 2 times negative 1 to the third power well negative 1 of the third power is just negative 1 times 2 so this is negative 2 and G prime of X I'll do it here G prime of X which use the power rule bring that 3 out front 3 times 2 is 6x decrement that exponent 3 minus 1 is 2 and so G prime of negative 1 is equal to 6 times negative 1 squared well negative 1 squared is just 1 so this is going to be equal to 6 so we actually know what all of these values are now we know so first we want to figure out F prime of negative 1 well they tell us that right over your F prime of negative 1 is equal to 5 so that is 5 G of negative 1 well we figure that right here G of negative 1 is negative 2 so this is negative 2 F of negative 1 V so F of negative 1 they tell us that right over there that is equal to 3 and then G prime of negative 1 let me just circle it in this green color G prime of negative 1 we figured it out it is equal to 6 so this is equal to 6 and then finally G of negative 1 right over here we already figure that out that was equal to negative 2 so negative 2 so this is all going to simplify to so you have 5 times negative 2 which is negative 10 minus 3 times 6 which is 18 all of that over negative 2 squared well negative 2 squared is just going to be positive 4 so this is going to be equal to negative 28 over positive 4 which is equal to negative 7 and there you have it it looks intimidating at first but if you just say okay look this is I can use them I can use the quotient rule right over here and then once I apply the quotient rule I can actually just directly figure out what G of negative 1 G prime of negative 1 and they gave us F of negative 1 and F prime of negative 1 so hopefully find that helpful