Worked example: Product rule with mixed implicit & explicit
Given the values of f and f' at x=-1 and that g(x)=1/x, Sal evaluates the derivative of F(x)=f(x)⋅g(x) at x=-1.
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- Since we have g(-1) = -1, so why doesn't taking derivative of a constant i.e g'(-1) = 0? Where am I wrong?(1 vote)
- Just because g(-1) = -1 does not make g a constant function! All g(-1) = -1 tells us is that g has value -1 specifically when x is -1.(15 votes)
- As for the title of this video, "Product rule with mixed implicit & explicit", is the function f implicit?(4 votes)
- In math, an explicit function is simply a function where the dependent variable is given explicitly; you don't have to algebraically manipulate the function to know what the dependent variable is. Every y=f(x) is an explicit function because it is clear that the value of y is dependent on the value of x.
On the other side, an implicit function is any "function" where there doesn't appear to be any dependent variable, such as x^2+y^2=1. Later on, you will likely learn about implicit differentiation, in which you calculate the slope of a curve given by any an implicit function, rather than just taking the derivative of an explicit function like we are doing currently.
It appears that neither f nor g is defined implicitly; they're both explicit, so the title of this video isn't helpful.(7 votes)
- The product rule is very similar to the sum identity for trignometry: Sin(x+y)= Sin(x)Cos(y) + Cos(X)Sin(Y) . Do these identities/rules have anything in common?(1 vote)
- Hello. I'm not sure what I'm missing here, but doesn't -1(-1^-2) equal 1. Sal got -1. I am really putting myself out there challenging Sal, so can someone let me know if I am correct or incorrect.(0 votes)
- (-1)⁻²=1/(-1)²=1/1=1. Then the leftmost -1 out front leaves us with -1 overall.(1 vote)
- [Voiceover] Let F be a function such that F of negative one is three. And F prime of negative one is equal to five. Let G be the function G of X is equal to one over X. Let capital F be a function defined as the product of those other two functions. What is capital F prime of negative one? Well, we can just apply the product rule here, let me just rewrite let me just essentially state the product rule. Capital F prime of X is going to be equal to since capital F of X is the product of these two functions, when we apply the product rule this is gonna be F prime of X times G of X plus plus F of X times G prime of X. And so if we want to evaluate this at F of negative one capital F prime at negative one is equal to F prime of negative one times G of negative one plus function F evaluated at negative one times the derivative of G evaluated at negative one. And let's see if we can figure these things out. So do they tell us this anywhere? Can we figure this out? F prime of negative one. Well they tell us right over here F prime of negative one is equal to five. So this is equal to five. Now, let's actually stick with F. What is F of negative one? Well, they tell that to us right over here. F of negative one is equal to three. So F of negative one is equal to three. Now G of negative one and G prime of negative one they don't give it to us explicitly here but we can figure it out. We can, we know that if G of X is equal to this G of negative one is equal to one over negative one which is equal to negative one. So this is equal to negative one. And then last but not least if we want to find G prime of negative one we just have to take the derivative of this. So G prime of X. Actually just let me rewrite G of X. G of X one over X is just the same thing as X to the negative one. So we're gonna use a power rule to figure out G prime of X is equal to bring that exponent out front, negative one times X to the and then decrement the exponent negative two power. So, G prime of negative one of negative one is equal to negative one times negative one to the negative two power. And that's just the same thing as negative one over negative one squared. This is one. So this is just all going to evaluate to negative one. So this is negative one. And so we have five times negative one which is negative five. Plus three times negative one which is negative three. Which is equal to negative eight. So F prime of negative one is equal to negative eight.