How do you find the derivative of $y = e^{2 x^{2} + 2 x}$ $y=e^(2x^2+2x)$ ?

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How do you find the derivative of $y = e^{2 x^{2} + 2 x}$ $y=e^(2x^2+2x)$ ?

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Answer 1 · 2018-04-14T22:27:21

$\frac{d}{d x} e^{2 x^{2} + 2 x} = (4 x + 2) e^{2 x^{2} + 2 x}$ $d/dxe^(2x^2+2x)=(4x+2)e^(2x^2+2x)$

Explanation:

This problem will require an application of the Chain Rule which, when applied to $e$ $e$ raised to the power of some function $u,$ $u,$ tells us that

$\frac{d}{d x} e^{u} = e^{u} \cdot \frac{d u}{d x}$ $d/dxe^u=e^u*(du)/dx$

Here, $u = 2 x^{2} + 2 x,$ $u=2x^2+2x,$ so we have

$\frac{d}{d x} e^{2 x^{2} + 2 x} = e^{2 x^{2} + 2 x} \cdot \frac{d}{d x} (2 x^{2} + 2 x)$ $d/dxe^(2x^2+2x)=e^(2x^2+2x)*d/dx(2x^2+2x)$

$\frac{d}{d x} (2 x^{2} + 2 x) = 4 x + 2$ $d/dx(2x^2+2x)=4x+2$ , so we end up with

$\frac{d}{d x} e^{2 x^{2} + 2 x} = (4 x + 2) e^{2 x^{2} + 2 x}$ $d/dxe^(2x^2+2x)=(4x+2)e^(2x^2+2x)$

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