00:01
In this video, we'll be using the product rule.
00:06
What is the product rule again? well, imagine we have a function, which we're trying to differentiate, and it can be written as the product of two functions, f and g.
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Let's call them.
00:21
And so if we want to differentiate this function, which we can also give a name, let's say h, then the product rule tells us how to do this.
00:36
So the product rule says that for any product of function, functions h, the derivative of h is the derivative of f times g, plus f times the derivative of g.
00:56
And we can write in the x's if we want, but the point is that we have the functions and derivatives of functions.
01:06
So f prime times g plus f times g prime.
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So it's like taking this, the h, and adding it to itself twice, except instead of fg here, we should have f prime, instead of fg here we should have g prime, fg prime.
01:33
So kind of like copying and pasting f times g twice and then differentiating f here and differentiating g here.
01:42
Okay, so now let's try using this.
01:48
If we have the function x times x minus 4, then the derivative of this function, not equal to, but once we differentiate, then we would get the function that is this first function differentiated multiplied by the second function.
02:13
So the derivative of x is 1.
02:18
So that part is 1 times x minus 4.
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And then our second term is this function times the derivative of this function, which is also 1 in this case.
02:34
And so the derivative of our product function here is 2x minus 4.
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So remember, the derivative of a function times another function is an expression with two terms being added.
02:54
The first term has the first function here differentiated, and that multiplied by the second function.
03:03
And the second term here has the first term, not differentiated, just the first function, multiplied by the second differentiated, the derivative of the second.
03:16
Okay, so let's try another example.
03:22
If we have the function x squared times 2x minus 1, the derivative of that function will be the derivative, of x squared multiplied by 2x minus 1 plus x squared times the derivative of 2x minus 1.
03:43
And now we can simplify that by expanding this out and adding this, and there we have that function.
04:07
Okay, so we have four more examples here, so let's go through them.
04:15
The derivative of this function here is the derivative of this function times this function.
04:21
So prime usually represents differentiation and then plus this function times the derivative of this function so we know the derivative of 3x plus 2 is just 3 since that's the derivative of 3x and then the derivative of 2 is 0 so the the derivative of the first 1st function times the second plus the first function times the derivative of the second.
05:07
And we can expand those to simplify.
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The main step of simplifying is adding like terms.
05:24
For this derivative, we get 12x minus 17.
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And this product rule is useful because if we didn't have it, then we would have to expand this first and then differentiate that.
05:45
In this case, that might not be too complicated.
05:49
But when we have really complicated products of functions, it could get pretty tricky, and the product rule simplifies things a lot.
06:01
Next example, we have the function 5x to the power of 7 plus 1, multiplied by the function x squared minus 2x.
06:14
Now notice we can factor out an x in this function and write this as a product of 3 functions, and then, in fact, we can use the product rule for three functions, but that's a topic for another video.
06:29
For now, we're just using the product rule for two functions.
06:35
So to differentiate this function, let's name the function b of x, so we can just say the derivative is b prime of x.
06:44
Now b prime of x is going to be the derivative of this times this plus the derivative of this times this.
07:04
So for now i've just written the parts that are the derivatives of these functions, and now i will write in the rest.
07:22
Now we want to simplify this as usual.
07:49
Now we move on to this fifth example.
07:55
This time i'm going to say y is a function of t, and then write the derivative of y with respect to t like this.
08:07
One might also write y.
08:09
Of t the function like this and then it's derivative as y prime of t that also works now let's find the derivative it's going to be the first function multiple or sorry the derivative of the first function times the second plus the first function times the derivative of the second and the derivative of the second is easy to differentiate since it's a polynomial and we already know how to to differentiate polynomials...