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

In summary, the conversation was about solving an exercise involving finding the derivative of a function. The participant made an attempt but encountered a mistake and sought help. After receiving assistance, the correct answer was obtained. The conversation also highlighted the limitations of rigid online homework apps.
  • #1
lastochka
29
0
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!
 
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  • #2
Here is what I get:

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

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

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

\(\displaystyle f'(x)=8x\left(2x^2-3\right)^{3}\left(9x^2+7\right)^{11}\left(72x^2-67\right)\)
 
  • #3
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...
 
  • #4
lastochka said:
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.
 
  • #5
Thank you!
 
  • #6
Perhaps the app is expecting:

\(\displaystyle f'(x)=8x\left(-2x^2+3\right)^{3}\left(9x^2+7\right)^{11}\left(-72x^2+67\right)\)
 
  • #7
MarkFL said:
Perhaps the app is expecting:

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

Yes, this is it! Thank you so much!
 

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

What is the chain rule in calculus?

The chain rule is a fundamental rule in calculus that allows you to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

How do I apply the chain rule?

To apply the chain rule, you need to identify the composite function and break it down into the outer function and inner function. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. Be sure to use the chain rule notation, which is d/dx.

Why is the chain rule important?

The chain rule is important because it allows us to find the derivatives of complex functions by breaking them down into simpler functions. This is useful in many applications, such as optimization problems and physics equations.

Can you give an example of using the chain rule?

Sure, let's say we have the function f(x) = (x^2 + 3)^4. To find the derivative, we can use the chain rule to break it down into the outer function g(x) = x^4 and the inner function h(x) = x^2 + 3. Then, using the chain rule, we get f'(x) = g'(h(x)) * h'(x) = 4(x^2 + 3)^3 * 2x = 8x(x^2 + 3)^3.

Are there any common mistakes when using the chain rule?

Yes, some common mistakes when using the chain rule include forgetting to use the chain rule notation, not identifying the composite function correctly, and making errors in the derivatives of the individual functions. It's important to carefully follow the steps and notation when using the chain rule to avoid these mistakes.

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