How do you find the derivative of $f (x) = e^{\sqrt{x}}$ $f(x)=e^(sqrtx)$ ?

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Answer 1 · 2018-02-15T14:02:17

$\frac{d}{d x} (e^{\sqrt{x}}) = \frac{e^{\sqrt{x}}}{2 \sqrt{x}}$ $d/dx (e^sqrtx) = e^sqrtx/(2sqrtx)$

Using the chain rule:

$\frac{d}{d x} f (y (x)) = \frac{d f}{d y} \times \frac{d y}{d x}$ $d/dx f(y(x)) = (df)/dy xx dy/dx$

with $y = \sqrt{x}$ $y=sqrtx$ :

$\frac{d}{d x} (e^{\sqrt{x}}) = (\frac{d}{d y} e^{y}) (\frac{d}{d x} \sqrt{x})$ $d/dx (e^sqrtx) = (d/dy e^y) (d/dx sqrt x)$

$\frac{d}{d x} (e^{\sqrt{x}}) = \frac{e^{y}}{2 \sqrt{x}}$ $d/dx (e^sqrtx) = e^y/(2sqrtx)$

$\frac{d}{d x} (e^{\sqrt{x}}) = \frac{e^{\sqrt{x}}}{2 \sqrt{x}}$ $d/dx (e^sqrtx) = e^sqrtx/(2sqrtx)$

How do you find the derivative of f(x)=e√xf(x)=e^(sqrtx)?