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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

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Final Answer

$\frac{\cos\left(x\right)\left(1+\cos\left(x\right)\right)+\sin\left(x\right)^2}{\left(1+\cos\left(x\right)\right)^2}+\frac{\ln\left(1+x^2\right)}{1+x^2}+\frac{2x\arctan\left(x\right)}{1+x^2}$
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Step-by-step Solution

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

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

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

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Learn how to solve 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}}.

Final Answer

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

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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 derivative of (sinx/(1+cosx)+arctanxln(1+x^2)) using the product ruleFind derivative of (sinx/(1+cosx)+arctanxln(1+x^2)) using the quotient ruleFind derivative of (sinx/(1+cosx)+arctanxln(1+x^2)) using logarithmic differentiation

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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

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