Mastering the Chain Rule: Derivative of a Complex Function | Tips & Tricks

[lastochka]

Hello,
I solved this exercise, but I probably did mistake in simplification...
f(x)=${\left(-2{x}^{2}+3\right)}^{4}$${\left(9{x}^{2}+7\right)}^{12}$
They asked to find derivative, so here is what I did
f$^{\prime}$=4${\left(-2{x}^{2}+3\right)}^{3}$(-4x)${\left(9{x}^{2}+7\right)}^{12}$+${\left(-2{x}^{2}+3\right)}^{4}$(12)${\left(9{x}^{2}+7\right)}^{11}$(18x)=
${\left(-2{x}^{2}+3\right)}^{3}$${\left(9{x}^{2}+7\right)}^{11}$(-504${x}^{3}$+592x)
There is a mistake and I can't find it. I did this exercise twice with the same result. I will appreciate your help, thanks!

 

[MarkFL]

Here is what I get:

$$f(x)=\left(-2x^2+3\right)^{4}\left(9x^2+7\right)^{12}$$

$$f'(x)=4\left(-2x^2+3\right)^{3}(-4x)\left(9x^2+7\right)^{12}+\left(-2x^2+3\right)^{4}\left(12\left(9x^2+7\right)^{11}(18x)\right)$$

$$f'(x)=8x\left(-2x^2+3\right)^{3}\left(9x^2+7\right)^{11}\left(-2\left(9x^2+7\right)+\left(-2x^2+3\right)\left(27\right)\right)$$

$$f'(x)=8x\left(2x^2-3\right)^{3}\left(9x^2+7\right)^{11}\left(72x^2-67\right)$$

 

[lastochka]

Thank you! I tried your answer, but it is still wrong according to my school exercise system (we have online tool for practice questions), may be it is just a format it doesn't accept...
By the way, I am so bad at algebra, I am still confused about your simplification... where -2 and 27 came from? Sorry, I am so bad at this...

 

[MarkFL]

Thank you! I tried your answer, but it is still wrong according to my school exercise system (we have online tool for practice questions), may be it is just a format it doesn't accept...
By the way, I am so bad at algebra, I am still confused about your simplification... where -2 and 27 came from? Sorry, I am so bad at this...

I really loathe those rigid online homework apps that will accept only 1 defined answer, when there are so many ways to write a result in many cases.

When I factored out the $8x$, then $-16x$ became $-2$ and $216x$ became $27$.

Also I factored out some negatives so that no factor would have a leading negative.

 

[lastochka]

Thank you!

 

[MarkFL]

Perhaps the app is expecting:

$$f'(x)=8x\left(-2x^2+3\right)^{3}\left(9x^2+7\right)^{11}\left(-72x^2+67\right)$$

 

[lastochka]

Perhaps the app is expecting:

$$f'(x)=8x\left(-2x^2+3\right)^{3}\left(9x^2+7\right)^{11}\left(-72x^2+67\right)$$

Yes, this is it! Thank you so much!