Final Answer
$\frac{\cos\left(\ln\left(x+1\right)\right)}{x+1}$
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Step-by-step Solution
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1
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 ${f'(x) = \cos(x)\cdot D_x(x)}$
$\frac{d}{dx}\left(\ln\left(x+1\right)\right)\cos\left(\ln\left(x+1\right)\right)$
2
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{x+1}\frac{d}{dx}\left(x+1\right)\cos\left(\ln\left(x+1\right)\right)$
3
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{1}{x+1}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)\cos\left(\ln\left(x+1\right)\right)$
4
The derivative of the constant function ($1$) is equal to zero
$\frac{1}{x+1}\frac{d}{dx}\left(x\right)\cos\left(\ln\left(x+1\right)\right)$
5
The derivative of the linear function is equal to $1$
$\frac{1}{x+1}\cos\left(\ln\left(x+1\right)\right)$
6
Multiply the fraction and term
$\frac{\cos\left(\ln\left(x+1\right)\right)}{x+1}$
Final Answer
$\frac{\cos\left(\ln\left(x+1\right)\right)}{x+1}$