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I need help to find the derivative of $(9x^6+4x^3)^4 $

I already tried the chain rule but I got the wrong answer and I do not know what I did wrong.

This is what I got:

$$4(9x^6+4x^3)^3 \times 54x^5+12x^2$$

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    Hello, The best thing is to show us what you did. We can point out the mistake. That's how we learn – 2017-02-15
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    Neither do we, we can't read above your shoulder. – 2017-02-15
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    4(9x^6+4x^3)^3 x 54x^5+12x^2 this is what I got – 2017-02-15
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    @DavidRomero You are correct. – 2017-02-15
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    @DavidRomero Apart from the missed brackets, do you have to type the answer in the computer, is it online hw ? Maybe the formatting is the issue – 2017-02-15
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    David Romero: In the future please add things such as the answer you got to **the question body**. Not everybody will see it in a comment. I added it for you this time. – 2017-02-15
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    @S.C.B. Why did you add the parens to David's answer? He didn't have them, and that may exactly have been the error. – 2017-02-15
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    @JyrkiLahtonen Hmm..I didn't see the comment. I thought you made a typo. – 2017-02-15
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    You are just missing parenthesis. – 2017-02-15

4 Answers 4

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$$f(g)=g^4,\\ g(x)=9x^6+4x^3.$$

Then

$$\frac{df(g)}{dg}=4g^3,\\\frac{dg(x)}{dx}=54x^5+12x^2$$ and

$$\frac{df(g(x))}{dx}=4g^3(x)(6x^5+12x^2).$$

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using the chain and the power rule we get $$4(9x^6+4x^3)^4(54x^5+12x^2)$$ simplifying we obtain $$24x^{11}(9x^3+2)(9x^3+4)^3$$

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Your answer is merely missing a pair of parentheses around the $54x^5+12x^2$. If you weren't already aware how important parentheses can be, hopefully you are now.

I'm guessing this was an exercise done at a computer, and the computer only told you you got it wrong, without any further explanation. I can't imagine a human grader with a red pen not inking in the correction, even if they are hardnosed and give no partial credit for the attempt.

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Apply the chain rule: $\frac{df(y)}{dx}=\frac{df}{dy}\frac{dy}{dx}$.

Let $y=(9x^6+4x^4)$. So you get $\frac{d}{dy}(y^4)=4y^3$ and $\frac{d}{dx}(9x^6+4x^4)=54x^5+12x^2$.

So, $\frac{df}{dy}\frac{dy}{dx}=4y^3(54x^5+12x^2)$. Substitute $y$ back with $(9x^6+4x^4)$ and you get:

$4(9x^6+4x^4)^3(54x^5+12x^2)$