What is the derivative of e^{2x}sqrt(x+1) ?

The derivative of e^{2x}sqrt(x+1) is (4e^{2x}x+5e^{2x})/(2sqrt(x+1))

What is the first derivative of e^{2x}sqrt(x+1) ?

The first derivative of e^{2x}sqrt(x+1) is (4e^{2x}x+5e^{2x})/(2sqrt(x+1))

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\frac{d}{d x} (e^{2 x} x + 1)

\frac{4 e ^{2 x} x + 5 e ^{2 x}}{2 x + 1}

Solution steps

\frac{d}{d x} (e^{2 x} x + 1)

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{2 x}, g = x + 1

= \frac{d}{d x} (e^{2 x}) x + 1 + \frac{d}{d x} (x + 1) e^{2 x}

\frac{d}{d x} (e^{2 x}) = e^{2 x} \cdot 2

\frac{d}{d x} (x + 1) = \frac{1}{2 x + 1}

= e^{2 x} \cdot 2 x + 1 + \frac{1}{2 x + 1} e^{2 x}

Simplify

e^{2 x} \cdot 2 x + 1 + \frac{1}{2 x + 1} e^{2 x} : \frac{4 e ^{2 x} x + 5 e ^{2 x}}{2 x + 1}

= \frac{4 e ^{2 x} x + 5 e ^{2 x}}{2 x + 1}

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What is the derivative of e^{2x}sqrt(x+1) ?
The derivative of e^{2x}sqrt(x+1) is (4e^{2x}x+5e^{2x})/(2sqrt(x+1))
What is the first derivative of e^{2x}sqrt(x+1) ?
The first derivative of e^{2x}sqrt(x+1) is (4e^{2x}x+5e^{2x})/(2sqrt(x+1))