Final Answer
$\sqrt{x+1}\left(2+x\right)$
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Step-by-step Solution
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$\frac{d}{dx}\left(\frac{2}{5}\sqrt{\left(x+1\right)^{5}}+\frac{2}{3}\sqrt{\left(x+1\right)^{3}}\right)$
2
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{d}{dx}\left(\frac{2}{5}\sqrt{\left(x+1\right)^{5}}\right)+\frac{d}{dx}\left(\frac{2}{3}\sqrt{\left(x+1\right)^{3}}\right)$
Intermediate steps
3
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$\frac{2}{5}\frac{d}{dx}\left(\sqrt{\left(x+1\right)^{5}}\right)+\frac{2}{3}\frac{d}{dx}\left(\sqrt{\left(x+1\right)^{3}}\right)$
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4
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$1\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x+1\right)+\frac{2}{3}\frac{d}{dx}\left(\sqrt{\left(x+1\right)^{3}}\right)$
5
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$1\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x+1\right)+1\sqrt{x+1}\frac{d}{dx}\left(x+1\right)$
6
Any expression multiplied by $1$ is equal to itself
$\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x+1\right)+1\sqrt{x+1}\frac{d}{dx}\left(x+1\right)$
7
Any expression multiplied by $1$ is equal to itself
$\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x+1\right)+\sqrt{x+1}\frac{d}{dx}\left(x+1\right)$
8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\sqrt{\left(x+1\right)^{3}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)+\sqrt{x+1}\frac{d}{dx}\left(x+1\right)$
9
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\sqrt{\left(x+1\right)^{3}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)+\sqrt{x+1}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)$
10
The derivative of the constant function ($1$) is equal to zero
$\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x\right)+\sqrt{x+1}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)$
11
The derivative of the constant function ($1$) is equal to zero
$\sqrt{\left(x+1\right)^{3}}\frac{d}{dx}\left(x\right)+\sqrt{x+1}\frac{d}{dx}\left(x\right)$
Intermediate steps
12
The derivative of the linear function is equal to $1$
$\sqrt{\left(x+1\right)^{3}}+\sqrt{x+1}\frac{d}{dx}\left(x\right)$
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Intermediate steps
13
The derivative of the linear function is equal to $1$
$\sqrt{\left(x+1\right)^{3}}+\sqrt{x+1}$
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Intermediate steps
14
Simplify the derivative
$\sqrt{x+1}\left(2+x\right)$
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Final Answer
$\sqrt{x+1}\left(2+x\right)$