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

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

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

## Step by step solution

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 $g=\sin\left(\cos\left(x\right)\right)$

$\frac{d}{dx}\left(\sin\left(\cos\left(x\right)\right)\right)\ln\left(\sin\left(x\right)\right)+\sin\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$
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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\ln\left(\sin\left(x\right)\right)\frac{d}{dx}\left(\cos\left(x\right)\right)\cos\left(\cos\left(x\right)\right)+\sin\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$
3

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

$\sin\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$
4

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\sin\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\sin\left(x\right)\right)\frac{1}{\sin\left(x\right)}-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$
5

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\sin\left(\cos\left(x\right)\right)\cos\left(x\right)\frac{1}{\sin\left(x\right)}-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$
6

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

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

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

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### Main topic:

Differential calculus

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