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Step-by-step Solution

Find the derivative using logarithmic differentiation method $\frac{d}{dx}\left(\left(2\sin\left(x\right)-1\cdot 3\cos\left(x\right)\right)^3\right)$

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Answer

$-108\sin\left(x\right)^2\cos\left(x\right)+54\cos\left(x\right)^{3}-72\cos\left(x\right)^2\sin\left(x\right)+36\sin\left(x\right)^{3}+24\sin\left(x\right)^{2}\cos\left(x\right)+81\cos\left(x\right)^{2}\sin\left(x\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\left(2\sin\left(x\right)-1\cdot 3\cdot \cos\left(x\right)\right)^3\right)$
1

Apply the rule of the cube of a binomial

$\frac{d}{dx}\left(\left(2\sin\left(x\right)-3\cos\left(x\right)\right)\left(4\sin\left(x\right)^2-12\sin\left(x\right)\cos\left(x\right)+9\cos\left(x\right)^2\right)\right)$
2

Multiplying polynomials $\cos\left(x\right)^2$ and $9\cdot 2\sin\left(x\right)+9-3\cos\left(x\right)$

$\frac{d}{dx}\left(8\sin\left(x\right)^{3}-12\sin\left(x\right)^2\cos\left(x\right)-24\sin\left(x\right)^2\cos\left(x\right)+36\cos\left(x\right)^2\sin\left(x\right)+18\cos\left(x\right)^2\sin\left(x\right)-27\cos\left(x\right)^{3}\right)$

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Answer

$-108\sin\left(x\right)^2\cos\left(x\right)+54\cos\left(x\right)^{3}-72\cos\left(x\right)^2\sin\left(x\right)+36\sin\left(x\right)^{3}+24\sin\left(x\right)^{2}\cos\left(x\right)+81\cos\left(x\right)^{2}\sin\left(x\right)$
$\frac{d}{dx}\left(\left(2\sin\left(x\right)-1\cdot 3\cdot \cos\left(x\right)\right)^3\right)$

Main topic:

Logarithmic differentiation

Related formulas:

6. See formulas

Time to solve it:

~ 0.47 seconds