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Lets take a look at question number seven. We are to find f prime. Prime.
If f is the absolute value of three x plus. Two okay so there possible solutions you can have for this the derivative of this function. So the first one.
Im going to be using the differentiation rule ddx of the absolute value you equals you over the absolute value. You the you the x okay. This is one possibly of finding the derivative of this function.
Okay so solution one were going to be using this formula right here. We also going to be applying the chain rule. Okay.
This is a composition of two functions is a composition of the absolute value function and the linear function. Okay so we have all f f of g of x equals the absolute value of three x plus two okay and we know what the chain rule is right so one of find the derivative of f of g of x. So f of g of x.
Prime is equal to f. Prime of g of x. The derivative of the outer function and i evaluated at the inner function times the derivative of the inner function.
So we are going to apply the chain rule here so lets see what the have we have all the compose this composite function take out f f. Of x. Is the absolute value x.
That is the outer function. Now the inner function is g of x. G.
Of x. Is of three x plus. Two okay.
Now is going to proceed to find the derivative of these two functions now the derivative of f f. Prime of x is x over the absolute value of x. The derivative of g g.
Prime of x is three. So what we going to do is find f. Prime of x.
And i waited at g.
So we just simply take g and plug it into the x is the f prime of x. Okay. And then multiply.
It by g prime of x. So f. Prime of x.
Is simply going to be f prime of g of x. Is instead of x divided by the absolute value x. Within a have three x is to taking the place of x.
So have three x plus. Two divided by the absolute value three x. Plus.
Two using the chain rule formula. We are to multiply this result by the derivative of the inner function g. Prime of x.
Okay so of g prime of x is three so just multiply. This by three. So thats your answer.
Okay. We can simplify into f. Prime.
Of x. Equals. Nine.
X. Plus. Six.
Divided by the absolute value of three x. Plus. Two.
So. There is your first solution now the second solution that we going to generate involves expressing this function is piecewise be find function okay on this result is also equally correct so solution to solution to were going to write the absolute value function as follows. This is going to be we know that f of x.
Which equals the absolute value three x.
This to can be written as plus or minus. Three x. Plus.
Two the sign change happens when the argument of the absolute value function changes sign okay so we just have to so three x was to equals zero. So this four x and that will tell you where you sign change. Happens okay.
So we subtract. Two and divide by three. We have x equals negative.
Two thirds prime. So now all function is negative three x. Plus.
Two. When x is less than negative. Two thirds and three x plus.
Two. When x is greater than or equal to two negative two thirds okay so now. This is our piece wise find function.
Now what were going to do is find the derivative of this piecewise be find function. So f prime of x is simply going to be the derivative of the function to the left. Which is negative three x.
Plus. Two the derivative of that is negative t. Three.
When x is less than negative to third and this on the bottom is three. When x is going to then or equal to negative to third. Okay.
So this is another possible solution for the derivative of the absolute value of three x plus. Two okay so any of these solutions you will get for credit you have to find the derivative of these absolute value function. Thanks so much for taking the time to watch this presentation.
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