differentiation question

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• January 7th 2009, 10:05 AM
entrepreneurforum.co.uk
differentiation question
Hello, how would i go about solving this equation?

z = (6x^4 - 6x^-3 - 2x + 5)^-2

im a little confused with the ^-2 at the end

im only supposed to find it to the first derivative

Also how would i go about solving this one to the first derivative?

x^2 e^3x^2-4x^3-3x+8 (its all one power, e *to the power of 3x * then 3x to the power of 2 .......)

i have this

dy/dx = 2x e^6x - 12x^2 - 3
• January 7th 2009, 10:23 AM
galactus
Quote:

Hello, how would i go about solving this equation?

$z = (6x^{4} - 6x^{-3} - 2x + 5)^{-2}$

im a little confused with the ^-2 at the end

im only supposed to find it to the first derivative
Use the chain rule.

$-2(6x^{4}-6x^{-3}-2x+5)^{-3}(24x^{3}+18x^{-4}-2)$

Now, simplify.
• January 7th 2009, 10:29 AM
Opalg
Quote:

Originally Posted by entrepreneurforum.co.uk
Hello, how would i go about solving this equation?

z = (6x^4 - 6x^-3 - 2x + 5)^-2

im a little confused with the ^-2 at the end

im only supposed to find it to the first derivative

First, get the terminology right. You're not "solving an equation", you're differentiating a function. So the question should be "How would I go about differentiating this function?"

Use the chain rule. The function z is of the form $z=y^{-2}$, where $y = 6x^4 - 6x^-3 - 2x + 5$. The derivative of $y^{-2}$ is $-2y^{-3}$, and the chain rule says that you must multiply this by the derivative of y.

Quote:

Originally Posted by entrepreneurforum.co.uk
Also how would i go about solving this one to the first derivative?

x^2 e^3x^2-4x^3-3x+8 (its all one power, e *to the power of 3x * then 3x to the power of 2 .......)

i have this

dy/dx = 2x e^6x - 12x^2 - 3

You got the $-4x^3-3x+8$ part right, so let's concentrate on differentiating the $x^2 e^{3x^2}$ bit. That is a product of two components, $x^2$ times $e^{3x^2}$. To differentiate it, you must use the product rule. That is not what you have done. Look up what the product rule for differentiation says, and have another go at this.
• January 7th 2009, 12:14 PM
tom@ballooncalculus
And, just in case a picture helps...

The chain rule generally...

http://www.ballooncalculus.org/chain_rule.png

... in which straight continuous lines differentiate downwards with respect to x, and the straight dashed line with respect to the dashed balloon expression. In this case...

http://www.ballooncalculus.org/chainDiff1.png

Introducing also the product rule thus...

http://www.ballooncalculus.org/mhf8.gif

or thus...

http://www.ballooncalculus.org/mhf9.gif

and in this case, incorporating the chain rule, too...

http://www.ballooncalculus.org/prodDiff1.png

Don't integrate - balloontegrate! Balloon Calculus: worked examples from past papers
• January 8th 2009, 04:28 AM
entrepreneurforum.co.uk
Opalg
this is what i have now for the second question

(x^2) d(e^3x^2) + d(x^2) (e^3x^2)

= (3^2)(6e^5x) + (2x) (e^3x^2)

is this right?
• January 8th 2009, 05:01 AM
Opalg
Quote:

Originally Posted by entrepreneurforum.co.uk
Opalg
this is what i have now for the second question

(x^2) d(e^3x^2) + d(x^2) (e^3x^2)

= (3^2)(6e^5x) + (2x) (e^3x^2)

is this right?

That's a step in the right direction. You are now applying the product rule correctly, but there's still a problem with differentiating e^3x^2.

Part of the problem is that you are interpreting the ambiguous expression e^3x^2 as though it means (e^3x)^2 = e^(6x). But I think it is much more likely to mean e^(3x^2). There is a difference between $(e^{3x})^2$ and $e^{3x^2}$, and you need to make sure that you are using the right one.

In either case, you need to use the chain rule to differentiate $e^y$ (where y is either 6x or 3x^2). The answer that you have given above, 6e^(5x), is wrong in either case.
• January 8th 2009, 05:08 AM
entrepreneurforum.co.uk
galactus

is this right?

-2(6x^4 - 6x^-3 - 2x +5) ^-3 (24x^3 +18x^-4 -2)

-2(24x + 18x - 2x + 3)^-3

do the power signs cancel each other out?

also im a little confused on this question, when finding the stationary points

y = (3- x)^3

I'm able to do these type of questions, but its the same problem as im getting with the question i asked earlier, about the power signs, so far i've got to

3(3-x)^2 (-x) = 0

and im just getting confused on where to go from there, any help would be great.

• January 8th 2009, 05:19 AM
entrepreneurforum.co.uk
opalg it does mean (e)^(3x^2)

would it be something like...

3x^2 (e) ^(3x^2 -1)

im getting a little lost with this area of differentiation now, is this going in the right direction?
• January 8th 2009, 05:19 AM
entrepreneurforum.co.uk
also thanks to everyone whos helped so far, im getting it alot more than what i did get before this thread, thanks
• January 8th 2009, 08:48 AM
Opalg
Quote:

Originally Posted by entrepreneurforum.co.uk
it does mean (e)^(3x^2)

would it be something like...

3x^2 (e) ^(3x^2 -1)

im getting a little lost with this area of differentiation now, is this going in the right direction?

No, still not there. (Shake)

The derivative of $e^x$ is $e^x$, not $e^{x-1}$. And the derivative of 3x^2 is 6x. So by the chain rule the derivative of $e^{3x^2}$ is $6xe^{3x^2}$.
• January 9th 2009, 07:44 AM
entrepreneurforum.co.uk
ok im trying to do this problem, and im still getting lost

$3x^2 e$^ $(3x^2 -5x^2 - 3x + 6 - 52)$

where everything after e^ in brackets is to the power of e
• January 9th 2009, 10:45 AM
tom@ballooncalculus
• January 10th 2009, 05:06 AM
entrepreneurforum.co.uk
tom im not sure what your trying to tell me, i kind of have an idea, but i know how to differentiate equations, just really stuck when it comes to the e^x function, as far as i know you don't differentiate the exponential function? so what i have is

$6x$ $e$^ $3x^2-5x^2 - 3x + 6 -52$

and what i think you tom are trying to show me is the product rule, and im not sure it applies here? i might be wrong?
• January 10th 2009, 07:53 AM
tom@ballooncalculus
Quote:

Originally Posted by entrepreneurforum.co.uk
...as far as i know you don't differentiate the exponential function?

Well, you do - what you've noticed is that the derivative is of $e^x$ is $e^x$. However, differentiating e to the power of anything more
complicated than x is going to need the chain rule. So you'll have as your derivative e to that same (complicated) power but the
whole thing then multiplied by the derivative of that (complicated) power.

Quote:

Originally Posted by entrepreneurforum.co.uk
...and what i think you tom are trying to show me is the product rule, and im not sure it applies here?

It always does apply if you're differentiating a product, as here - i.e. the product of $3x^2$ and e to that horrible power.
Zooming out of the chain rule to show the product rule alone...

http://www.ballooncalculus.org/prodDiff3.png

Don't integrate - balloontegrate! Balloon Calculus: worked examples from past papers
• January 12th 2009, 06:39 AM
entrepreneurforum.co.uk
thanks tom im slowly getting it more and more!

what i have now is

(3x^2(e^6x-10x-3) + (6x)(e^3x^2-5x^2-3x+6-52)

right or wrong?
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