Uploaded by TheIntegralCALC on 12.02.2010

Transcript:

Hey, everybody.

Okay.

Another chain rule problem:

this one is f(x) =√(2x+1) .

Okay.

So, this

this chain rule problem,

we are

we’re going to illustrate chain rule by taking the derivative.

So, taking the derivative we say f’(x) equals…

The first thing that we’re going to want to do,

and I'm actually going to go ahead and put this step up here before I move forward to the derivative,

we’ve shown in other videos that you can change something from

we’ve shown in other videos that you can change something from

the square root of something to that thing times 1/2.

It’s the same thing. So, what we’re going to do is change this to (2x+1)^(1/2),

which is the same thing as the square root of (2x+1).

So, we’re just going to go ahead and change it so that

it makes it easier for us to, not only take the derivative, but also to illustrate chain rule.

So, now that we’ve converted the problem,

what we’re going to do to take the derivative is go ahead and move

this exponent, 1/2, out front here,

and then subtract 1 from the exponent. So,

we have 1/2 times (2x+1)^(-1/2),

because we did 1/2 minus 1 so we get -1/2 here.

So, that’s our first step.

Of course, since (2x+1) is more complicated than a simple variable,

we can’t just take the derivative on the outside of this without dealing with the inside.

What we have to do is, because of chain rule,

we have to multiply this function by the derivative of the inside here.

So, we take the derivative of 2x + 1.

2x + 1

The derivative here is just 2,

because that’s the derivative of 2x, and 1 is a constant so the derivative is 0, it goes away.

So, the derivative is 2, which means that we have to multiply this whole thing

by 2.

Now, we have not only addressed the outside of the function,

but we’ve also addressed the inside.

So, we have

we have the full problem,

the full derivative, and now all we have to do is simplify.

So, since we have 2 up here in the numerator and 2 in the denominator,

2 times 1/2 is 1.

These cancel, so these two goes away, the 1/2 goes away

and we end up with (2x+1)^(-1/2).

You could leave it this way but it’s usually not preferable

to leave a negative exponent in your answer.

So, what we would do is go ahead and change this

to 1 over the square root of (2x+1),

and I hope you guys can see it this low. But

But as

as we'd changed the square root over here into 1/2, the exponent, we can also change it back.

We have 1/2; we'd changed it back into a square root.

But because this exponent here is negative,

I have to move the square root to the bottom of the fraction to make it back into a positive.

So, I changed it back into a square root instead of 1/2

and I moved it to the bottom so that I could make this positive.

So, the final answer here is 1 over the square root of (2x+1).

See you guys later.

Okay.

Another chain rule problem:

this one is f(x) =√(2x+1) .

Okay.

So, this

this chain rule problem,

we are

we’re going to illustrate chain rule by taking the derivative.

So, taking the derivative we say f’(x) equals…

The first thing that we’re going to want to do,

and I'm actually going to go ahead and put this step up here before I move forward to the derivative,

we’ve shown in other videos that you can change something from

we’ve shown in other videos that you can change something from

the square root of something to that thing times 1/2.

It’s the same thing. So, what we’re going to do is change this to (2x+1)^(1/2),

which is the same thing as the square root of (2x+1).

So, we’re just going to go ahead and change it so that

it makes it easier for us to, not only take the derivative, but also to illustrate chain rule.

So, now that we’ve converted the problem,

what we’re going to do to take the derivative is go ahead and move

this exponent, 1/2, out front here,

and then subtract 1 from the exponent. So,

we have 1/2 times (2x+1)^(-1/2),

because we did 1/2 minus 1 so we get -1/2 here.

So, that’s our first step.

Of course, since (2x+1) is more complicated than a simple variable,

we can’t just take the derivative on the outside of this without dealing with the inside.

What we have to do is, because of chain rule,

we have to multiply this function by the derivative of the inside here.

So, we take the derivative of 2x + 1.

2x + 1

The derivative here is just 2,

because that’s the derivative of 2x, and 1 is a constant so the derivative is 0, it goes away.

So, the derivative is 2, which means that we have to multiply this whole thing

by 2.

Now, we have not only addressed the outside of the function,

but we’ve also addressed the inside.

So, we have

we have the full problem,

the full derivative, and now all we have to do is simplify.

So, since we have 2 up here in the numerator and 2 in the denominator,

2 times 1/2 is 1.

These cancel, so these two goes away, the 1/2 goes away

and we end up with (2x+1)^(-1/2).

You could leave it this way but it’s usually not preferable

to leave a negative exponent in your answer.

So, what we would do is go ahead and change this

to 1 over the square root of (2x+1),

and I hope you guys can see it this low. But

But as

as we'd changed the square root over here into 1/2, the exponent, we can also change it back.

We have 1/2; we'd changed it back into a square root.

But because this exponent here is negative,

I have to move the square root to the bottom of the fraction to make it back into a positive.

So, I changed it back into a square root instead of 1/2

and I moved it to the bottom so that I could make this positive.

So, the final answer here is 1 over the square root of (2x+1).

See you guys later.