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

Find the derivative of ln(sin(x))sin(cos(x))

Answer

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

Step-by-step explanation

Problem

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

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

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Answer

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

Main topic:

Differential calculus

Used formulas:

2. See formulas

Time to solve it:

~ 0.58 seconds