Find the derivative of (sin(x))/(1+cos(x))+arctan(x)ln(1+x^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)

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Answer

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

Step by step solution

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

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=arctan\left(x\right)$ and $g=\ln\left(x^2+1\right)$

$\frac{d}{dx}\left(\frac{\sin\left(x\right)}{\cos\left(x\right)+1}\right)+\frac{d}{dx}\left(\ln\left(x^2+1\right)\right)arctan\left(x\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
3

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{d}{dx}\left(\frac{\sin\left(x\right)}{\cos\left(x\right)+1}\right)+\frac{1}{x^2+1}arctan\left(x\right)\frac{d}{dx}\left(x^2+1\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
4

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\frac{d}{dx}\left(\cos\left(x\right)+1\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\frac{1}{x^2+1}arctan\left(x\right)\frac{d}{dx}\left(x^2+1\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
6

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

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(\frac{d}{dx}\left(\cos\left(x\right)\right)+\frac{d}{dx}\left(1\right)\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\frac{1}{x^2+1}arctan\left(x\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
7

The derivative of the constant function is equal to zero

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(\frac{d}{dx}\left(\cos\left(x\right)\right)+0\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\frac{1}{x^2+1}arctan\left(x\right)\left(\frac{d}{dx}\left(x^2\right)+0\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
8

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(\frac{d}{dx}\left(\cos\left(x\right)\right)+0\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\left(2x+0\right)\frac{1}{x^2+1}arctan\left(x\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
9

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 $f'(x) = -\sin(x)\cdot D_x(x)$

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(0-\sin\left(x\right)\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\left(2x+0\right)\frac{1}{x^2+1}arctan\left(x\right)+\ln\left(x^2+1\right)\frac{d}{dx}\left(arctan\left(x\right)\right)$
10

Taking the derivative of arctangent

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(0-\sin\left(x\right)\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\left(2x+0\right)\frac{1}{x^2+1}arctan\left(x\right)+\frac{1}{x^2+1}\ln\left(x^2+1\right)\frac{d}{dx}\left(x\right)$
11

The derivative of the linear function is equal to $1$

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-\left(0-\sin\left(x\right)\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+\left(2x+0\right)\frac{1}{x^2+1}arctan\left(x\right)+1\left(\frac{1}{x^2+1}\right)\ln\left(x^2+1\right)$
12

$x+0=x$, where $x$ is any expression

$\frac{\left(\cos\left(x\right)+1\right)\cos\left(x\right)-1\left(-1\right)\sin\left(x\right)\sin\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+2x\frac{1}{x^2+1}arctan\left(x\right)+1\left(\frac{1}{x^2+1}\right)\ln\left(x^2+1\right)$
13

Multiply $-1$ times $-1$

$\frac{1\sin\left(x\right)\sin\left(x\right)+\left(\cos\left(x\right)+1\right)\cos\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+2x\frac{1}{x^2+1}arctan\left(x\right)+1\left(\frac{1}{x^2+1}\right)\ln\left(x^2+1\right)$
14

Any expression multiplied by $1$ is equal to itself

$\frac{\sin\left(x\right)\sin\left(x\right)+\left(\cos\left(x\right)+1\right)\cos\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+2x\frac{1}{x^2+1}arctan\left(x\right)+\frac{1}{x^2+1}\ln\left(x^2+1\right)$
15

When multiplying exponents with same base you can add the exponents

$\frac{\sin\left(x\right)^2+\left(\cos\left(x\right)+1\right)\cos\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+2x\frac{1}{x^2+1}arctan\left(x\right)+\frac{1}{x^2+1}\ln\left(x^2+1\right)$
16

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=2$ and $x=x^2+1$

$\frac{\sin\left(x\right)^2+\left(\cos\left(x\right)+1\right)\cos\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+x\frac{2}{x^2+1}arctan\left(x\right)+\frac{1}{x^2+1}\ln\left(x^2+1\right)$
17

Using the power rule of logarithms

$\frac{\sin\left(x\right)^2+\left(\cos\left(x\right)+1\right)\cos\left(x\right)}{\left(\cos\left(x\right)+1\right)^2}+x\frac{2}{x^2+1}arctan\left(x\right)+\ln\left(\left(x^2+1\right)^{\left(\frac{1}{x^2+1}\right)}\right)$
18

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

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

Answer

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