If f of x equals the given function, we want to find the derivative function or f prime of
x and then find f prime of one. Notice how the function is given as a product of two functions,
so if we want to find the derivative function in this form, we'll have to apply the product
rule of differentiation given here below, where the derivative of function f times function g is
equal to f times g prime plus g times f prime, or we can say the first function times the derivative
of the second function plus the second function times the derivative of the first function.
Of course, the other option would be to find this product first and then find the
derivative function using the basic power rule. The main purpose is to introduce the
product rule. So, we'll go ahead and apply the product rule to find our derivative,
but toward the end of the video, we'll also show that if we multiply this out
and then find the derivative, the result would be the same.
Using our formula given here below, we'll let the first function be function f and the second
function be function g. Now we'll set up the product rule. So f prime of x is going to be equal
to f times g prime, or the first function times the derivative of the second function, plus g
times f prime, which should be the second function times the derivative of the first function.
So now we'll find the derivative here and here, then find the products and then simplify.
So f prime of x is going to be equal to again the first function times the derivative of three x
squared minus two. To find the derivative of three x squared, we're going to multiply by two, which
would give us six, subtract one from the exponent, which would be x to the first power, or just x,
minus the derivative of two, which would be zero. So our derivative is just six x.
Then we'll have three x squared minus two times the derivative of x squared minus five x plus
eight. The derivative of x squared would be two x, minus the derivative of five x, which would
just be five, and the derivative of eight would be zero. So our derivative is two x minus five.
So now we'll find these products here. We'll have three products: one, two, three, and here we'll
have four products: one, two, three, and four. So we have f prime of x equals six x times three
x squared minus thirty, which gives us thirty x squared plus forty-eight x. Now for this product,
we'll have plus six x times three
x squared minus fifteen x squared minus four x, and then finally plus ten.
Now we'll go ahead and combine like terms. There are two x cubed terms,
there are two x squared terms, and there are also two x terms. So f prime of x is equal to twelve
x cubed minus forty-five x squared plus forty-four x plus ten. So this is our derivative function.
To find f prime of one, which would be the slope of the tangent line at x equals one, we'll
substitute one for x into our derivative function. So we have twelve minus forty-five, which is
minus thirty-three, plus forty-four, which is plus eleven, plus ten. So f prime of one is
equal to twenty-one, which again will be the slope of the tangent line when x equals one.
So let's go and take a look at a graph. The blue graph is the graph of the original function when
x equals one would be at this point here on the function. The red line would be the
tangent line. Notice how the y-axis is scaled by fives, so the slope of this red tangent line
does appear to be positive twenty-one, which verifies that our work is correct.
Now before we go, I do want to go back and show how if we were to find the derivative by first
finding this product, the derivative would be the same as we found using the product rule.
I've already worked this out, but I do want to show it again. If we were to start with the
original function and multiply this out, we would have a total of six products: one, two, three,
four, five, and six, which will give us these terms here. If we combine like terms, it would
give us this function. This is the same as the original function, but we've expanded the product.
Now if we apply the product rule to find the derivative of the function in this form,
notice how the derivative function is still the same. We have f prime of x equals twelve x
cubed minus forty-five x squared plus forty-four x plus ten. But again, the
main idea for this video was to show how to use the product rule. I hope you found this helpful.