# Step-by-step Solution

## Find the derivative using the product rule $\frac{d}{dx}\left(\ln\left(x^2+2\right)^3\left(1-x^3\right)^4\right)$

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### Videos

$\ln\left(x^2+2\right)^{2}x\left(1-x^3\right)^4\frac{6}{x^2+2}-12\ln\left(x^2+2\right)^3\left(1-x^3\right)^{3}x^{2}$

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(ln\left(x^2+2\right)^3\left(1-x^3\right)^4\right)$
1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(x^2+2\right)^3$ and $g=\left(1-x^3\right)^4$

$\left(1-x^3\right)^4\frac{d}{dx}\left(\ln\left(x^2+2\right)^3\right)+\ln\left(x^2+2\right)^3\frac{d}{dx}\left(\left(1-x^3\right)^4\right)$
2

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$3\ln\left(x^2+2\right)^{2}\left(1-x^3\right)^4\frac{d}{dx}\left(\ln\left(x^2+2\right)\right)+4\ln\left(x^2+2\right)^3\left(1-x^3\right)^{3}\cdot\frac{d}{dx}\left(1-x^3\right)$

$\ln\left(x^2+2\right)^{2}x\left(1-x^3\right)^4\frac{6}{x^2+2}-12\ln\left(x^2+2\right)^3\left(1-x^3\right)^{3}x^{2}$
$\frac{d}{dx}\left(ln\left(x^2+2\right)^3\left(1-x^3\right)^4\right)$

Product rule

~ 0.62 seconds