Mertsj
Derivative question



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myininaya
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yes?

Mertsj
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dw:1328147554735:dw

anonymous
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ho ho ho

Mertsj
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Hi Santa

myininaya
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Let F be the antiderivative of
\[f(x)=\frac{u^21}{u^2+1}\]
Now we have
\[[g(x)]'=[F(x)_{2x}^{3x}]=[F(3x)F(2x)]'=(3x)'f(3x)(2x)'f(2x)\]

myininaya
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that chain rule stuff is the best! :)

myininaya
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so we have
\[3 \cdot \frac{(3x)^21}{(3x)^2+1}2 \cdot \frac{(2x)^21}{(2x)^2+1}\]

myininaya
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\[3 \cdot \frac{9x^21}{9x^2+1}2 \cdot \frac{4x^21}{4x^2+1}\]

anonymous
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\[\int _a^{2x}\frac{u^21}{u^2+1}du+\int_a^{3x}\frac{u^21}{u^2+1}du\]

anonymous
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dang did that take me a long time to type
now chain rule

myininaya
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yes it did satellite :)

anonymous
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i am tired, time for bed. gnight

myininaya
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me too
we are going to bed

Mertsj
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\[g'(x)=\frac{2(4x^21)}{4x^2+1}+\frac{3(9x^21)}{9x^2+1}\]

Mertsj
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Not yet, I hope.

Mertsj
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I don't get the 2 and the 3

Mertsj
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Maybe if I study it.

myininaya
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:do you understand what I did?

Mertsj
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Not quite.

Mertsj
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I thought the u^21/u^2+1 would be the derivative and all I had to do was plug in the interval endpoints.

myininaya
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\[g(x)=\int\limits_{2x}^{3x}f(u) du=F(u)_{2x}^{3x}=F(3x)F(2x)\]
Do you understand this part?

Mertsj
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Why isn't f(u) the g'(x)?

myininaya
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no \[g(x)=F(3x)F(2x)\]

myininaya
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g is the integral from 2x to 3x of f(u) with respect to u

myininaya
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now since we know that
\[g(x)=F(3x)F(2x)\]
we can differentiate both sides

Mertsj
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Ok. I get that and I understand what you wrote about F(3x)F(2x)

myininaya
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\[g'(x)=(3x)'f(3x)(2x)'f(2x)\]

myininaya
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i used chain rule for this

Mertsj
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One second.

myininaya
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\[g'(x)=3 f(3x)2 f(2x)\]
since (3x)'=3 and (2x)'=2

myininaya
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ok one sec granted

Mertsj
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so F(3x)=(3x)'f(x)?

Mertsj
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And what happened to F(x)?

myininaya
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well (F(3x))'=(3x)'f(3x) by chain rule

myininaya
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remember we differentiate both sides so we could find g'

Mertsj
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Yep. Ok. I think i can figure it out now. thank you very much and how did you get so smart?

myininaya
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i'm a little smart not much

myininaya
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well i'm not really smart lol

myininaya
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but thanks mert

Mertsj
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thanks to you and sleep well...

myininaya
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thanks i'm going to sleep for reals
i'm so tired
by the way satellite likes to do these types of problems another way
but i love my way

myininaya
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he likes to split the integral at a constant

myininaya
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and then use the other part of the fundamental thm of calculus

myininaya
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goodnight! :)

Mertsj
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Satellite is very smart also. Do you know him personally?