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

The derivative of a(e^{2x}+2e^{2x}x) is a(4e^{2x}x+4e^{2x})

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

The first derivative of a(e^{2x}+2e^{2x}x) is a(4e^{2x}x+4e^{2x})

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

a (4 e^{2 x} x + 4 e^{2 x})

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= a (e^{2 x} \cdot 2 + 2 (2 e^{2 x} x + e^{2 x}))

Expand

e^{2 x} \cdot 2 + 2 (2 e^{2 x} x + e^{2 x}) : 4 e^{2 x} x + 4 e^{2 x}

= a (4 e^{2 x} x + 4 e^{2 x})

What is the derivative of a(e^{2x}+2e^{2x}x) ?
The derivative of a(e^{2x}+2e^{2x}x) is a(4e^{2x}x+4e^{2x})
What is the first derivative of a(e^{2x}+2e^{2x}x) ?
The first derivative of a(e^{2x}+2e^{2x}x) is a(4e^{2x}x+4e^{2x})