Try NerdPal! Our new app on iOS and Android

# Find the derivative $\frac{d}{dx}\left(\frac{\sin\left(x\right)}{1+\cos\left(x\right)}+\arctan\left(x\right)\ln\left(1+x^2\right)\right)$ using the sum rule

## Step-by-step Solution

Go!
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Videos

$\frac{\cos\left(x\right)\left(1+\cos\left(x\right)\right)+\sin\left(x\right)^2}{\left(1+\cos\left(x\right)\right)^2}+\frac{\ln\left(1+x^2\right)}{1+x^2}+\frac{2x\arctan\left(x\right)}{1+x^2}$
Got another answer? Verify it here!

## Step-by-step Solution

Problem to solve:

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

Specify the solving method

1

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

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

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

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

Learn how to solve sum rule of differentiation problems step by step online.

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

Learn how to solve sum rule of differentiation problems step by step online. Find the derivative d/dx((sin(x)/(1+cos(x))+arctan(x)ln(1+x^2)) using the sum rule. The derivative of a sum of two or more functions is the sum of the derivatives of each function. 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(1+x^2\right). 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}. Apply the quotient rule for differentiation, 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}}.

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

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind d/dx((sin(x)/(1+cos(x))+arctan(x)ln(1+x^2)) using the product ruleFind d/dx((sin(x)/(1+cos(x))+arctan(x)ln(1+x^2)) using the quotient ruleFind d/dx((sin(x)/(1+cos(x))+arctan(x)ln(1+x^2)) using logarithmic differentiation
SnapXam A2

### beta Got a different answer? Verify it!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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

### Main topic:

Sum Rule of Differentiation

10. See formulas

~ 0.18 s