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# Find the derivative of $\sqrt{\frac{x\left(x+2\right)}{\left(2x+1\right)\left(5x+3\right)}}$

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##  Final answer to the problem

$\frac{\left(x+2+x\right)\left(2x+1\right)\left(5x+3\right)+\left(-x-2\right)x\left(2\left(5x+3\right)+5\left(2x+1\right)\right)}{2\left(2x+1\right)^2\left(5x+3\right)^2}\sqrt{\frac{\left(2x+1\right)\left(5x+3\right)}{x\left(x+2\right)}}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• 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
Can't find a method? Tell us so we can add it.
1

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

$\frac{1}{2}\left(\frac{x\left(x+2\right)}{\left(2x+1\right)\left(5x+3\right)}\right)^{-\frac{1}{2}}\frac{d}{dx}\left(\frac{x\left(x+2\right)}{\left(2x+1\right)\left(5x+3\right)}\right)$

Learn how to solve problems step by step online.

$\frac{1}{2}\left(\frac{x\left(x+2\right)}{\left(2x+1\right)\left(5x+3\right)}\right)^{-\frac{1}{2}}\frac{d}{dx}\left(\frac{x\left(x+2\right)}{\left(2x+1\right)\left(5x+3\right)}\right)$

Learn how to solve problems step by step online. Find the derivative of ((x(x+2))/((2x+1)(5x+3)))^(1/2). The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. Since the exponent is negative, we can invert the fraction. 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}}. The power of a product is equal to the product of it's factors raised to the same power.

##  Final answer to the problem

$\frac{\left(x+2+x\right)\left(2x+1\right)\left(5x+3\right)+\left(-x-2\right)x\left(2\left(5x+3\right)+5\left(2x+1\right)\right)}{2\left(2x+1\right)^2\left(5x+3\right)^2}\sqrt{\frac{\left(2x+1\right)\left(5x+3\right)}{x\left(x+2\right)}}$

##  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

SnapXam A2

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0
a
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x
y
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.
(◻)
+
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◻/◻
/
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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