 # Evaluate the following derivatives. d/(dx)((x+1)^(2x)) boitshupoO 2021-02-19 Answered
Evaluate the following derivatives. $\frac{d}{dx}\left({\left(x+1\right)}^{2x}\right)$
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Step 1
To Determine: Evaluate the following derivatives.
Given: we have a function ${\left(x+1\right)}^{2}x$
Explanation, we have a function $y={\left(x+1\right)}^{2}x$ we have to find the derivative of y now we will applying exponent rule now we will applying the chain rule
$\frac{d}{dx}\left({e}^{2x\mathrm{ln}\left(x+1\right)}\right)={e}^{2x\mathrm{ln}\left(x+1\right)}\frac{d}{dx}\left(2x\mathrm{ln}\left(x+1\right)\right)$
Step 2
Now we will use product rule
${e}^{2x\mathrm{ln}\left(x+1\right)}\frac{d}{dx}\left(2x\mathrm{ln}\left(x+1\right)\right)={e}^{2x\mathrm{ln}\left(x+1\right)}×2×\left[x\frac{d}{dx}\mathrm{ln}\left(x+1\right)+\mathrm{ln}\left(x+1\right)\frac{dx}{dx}\right]$
$=2{e}^{2x\mathrm{ln}\left(x+1\right)}\left[\frac{x}{x+1}+\mathrm{ln}\left(x+1\right)\right]$
$=2{\left(x+1\right)}^{2x}\left[\frac{x}{x+1}+\mathrm{ln}\left(x+1\right)\right]$

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