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Step-by-step Solution

Find the derivative using the product rule (d/dx)(ln(sin(x))sin(cos(x)))

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Answer

$\cos\left(x\right)\sin\left(\cos\left(x\right)\right)\csc\left(x\right)-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\cdot\sin\left(\cos\left(x\right)\right)\right)$
1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(\sin\left(x\right)\right)$ and $g=\sin\left(\cos\left(x\right)\right)$

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)\sin\left(\cos\left(x\right)\right)+\ln\left(\sin\left(x\right)\right)\frac{d}{dx}\left(\sin\left(\cos\left(x\right)\right)\right)$
2

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)\sin\left(\cos\left(x\right)\right)+\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\cos\left(x\right)\right)$

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Answer

$\cos\left(x\right)\sin\left(\cos\left(x\right)\right)\csc\left(x\right)-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$
$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\cdot\sin\left(\cos\left(x\right)\right)\right)$

Main topic:

Differential calculus

Used formulas:

2. See formulas

Time to solve it:

~ 1.38 seconds

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