Find the derivative of $\sqrt{x}\arctan\left(x\right)$

Step-by-step Solution

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Final answer to the problem

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

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  • Find the derivative using the definition
  • Find the derivative using the product rule
  • Find the derivative using the quotient rule
  • Find the derivative using logarithmic differentiation
  • Find the derivative
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
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1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\arctan\left(x\right)$ and $g=\sqrt{x}$

$\frac{d}{dx}\left(\arctan\left(x\right)\right)\sqrt{x}+\frac{d}{dx}\left(\sqrt{x}\right)\arctan\left(x\right)$

Learn how to solve integrals with radicals problems step by step online.

$\frac{d}{dx}\left(\arctan\left(x\right)\right)\sqrt{x}+\frac{d}{dx}\left(\sqrt{x}\right)\arctan\left(x\right)$

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Learn how to solve integrals with radicals problems step by step online. Find the derivative of arctan(x)x^(1/2). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\arctan\left(x\right) and g=\sqrt{x}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. Taking the derivative of arctangent. The derivative of the linear function is equal to 1.

Final answer to the problem

$\frac{\sqrt{x}}{1+x^2}+\frac{\arctan\left(x\right)}{2\sqrt{x}}$

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Function Plot

Plotting: $\frac{\sqrt{x}}{1+x^2}+\frac{\arctan\left(x\right)}{2\sqrt{x}}$

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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

How to improve your answer:

Main Topic: Integrals with Radicals

Integrals with radicals are those integrals that contain a radical (square root, cubic, etc.) in the numerator or denominator of the integral.

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