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\frac{d}{dx}\left(\frac{\sin\left(x\right)}{1+\cos\left(x\right)}+arctan\left(x\right)\cdot \ln\left(1+x^2\right)\right)

Derive the function (sin(x))/(1+cos(x))+arctan(x)ln(1+x^2) with respect to x

Answer

$\frac{\left(\cos\left(x\right)+1\right)^2\left(x^2+1\right)\ln\left(x^2+1\right)+\left(x^2+1\right)\left(\left(x^2+1\right)\left(\sin\left(x\right)^2+\left(\cos\left(x\right)+1\right)\cos\left(x\right)\right)+2\left(\cos\left(x\right)+1\right)^2xarctan\left(x\right)\right)}{\left(\cos\left(x\right)+1\right)^2\left(x^2+1\right)^2}$

Step-by-step explanation

Problem

$\frac{d}{dx}\left(\frac{\sin\left(x\right)}{1+\cos\left(x\right)}+arctan\left(x\right)\cdot \ln\left(1+x^2\right)\right)$
1

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(\ln\left(x^2+1\right)arctan\left(x\right)\right)+\frac{d}{dx}\left(\frac{\sin\left(x\right)}{\cos\left(x\right)+1}\right)$

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Answer

$\frac{\left(\cos\left(x\right)+1\right)^2\left(x^2+1\right)\ln\left(x^2+1\right)+\left(x^2+1\right)\left(\left(x^2+1\right)\left(\sin\left(x\right)^2+\left(\cos\left(x\right)+1\right)\cos\left(x\right)\right)+2\left(\cos\left(x\right)+1\right)^2xarctan\left(x\right)\right)}{\left(\cos\left(x\right)+1\right)^2\left(x^2+1\right)^2}$
$\frac{d}{dx}\left(\frac{\sin\left(x\right)}{1+\cos\left(x\right)}+arctan\left(x\right)\cdot \ln\left(1+x^2\right)\right)$

Main topic:

Differential calculus

Used formulas:

7. See formulas

Time to solve it:

~ 0.68 seconds