Working an example: Product rule with mixed implicit and explicit
About this video
We know f(-1) = 3 and f'(-1) = 5, and g(x) = 1/x. Capital F is the product f(x)g(x). The video uses the product rule, finds g(-1) = -1 and g'(-1) = -1 with the power rule, and gets F'(-1) = 5(-1) + 3(-1) = -8.
You'll be able to
- apply the product rule when one function is given by values and the other by a formula.
- rewrite 1/x as x^(-1) and differentiate it with the power rule.
- substitute values to evaluate a product rule derivative at a point.
Key words
- product rule
- F'(x) = f'(x)g(x) + f(x)g'(x) when F(x) = f(x)g(x).
- power rule
- The derivative of x^n is n x^(n-1), which also works for negative n.
- explicit function
- A function given by a formula, like g(x) = 1/x.
Transcript
- [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.
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