Finding the derivative for an exponential function.

Question

anonymous · Answer

$$e ^{\sqrt{x}}$$ is my function, and somehow I need to get $$\frac{e ^{\sqrt{x}} }{ 2\sqrt{x} }$$

anonymous · Answer

I can get to $$\frac{ e ^{\sqrt{x}} }{ \sqrt{x} }$$ but I don't know how to get that 2.

mathstudent55 · Answer

For u, a function of x, 

$\dfrac{d}{dx} e^u = e^u \cdot \dfrac{d}{dx} u$

mathstudent55 · Answer

Start like this:

$\large \dfrac{d}{dx} e^\sqrt x = e^\sqrt x \cdot \dfrac{d}{dx} \sqrt x$

mathstudent55 · Answer

Now realize that $\large \sqrt x = x^{\frac{1}{2}}$.
Take the derivative using the power rule.

anonymous · Answer

wouldn't the drivative of that be $$x ^{\frac{ -1 }{ 2 }}$$

mathstudent55 · Answer

Almost.
You are missing the 1/2 exponent you need to multiply by before subtracting 1 from the exponent.
$\large \dfrac{d}{dx} e^\sqrt x = e^\sqrt x \cdot \dfrac{d}{dx} \sqrt x$

$~~~~~~~~~~~~~~~ = e^\sqrt x \cdot \dfrac{d}{dx} x^{\frac{1}{2}}$

$~~~~~~~~~~~~~~~ = e^\sqrt x \cdot \dfrac{1}{2} x^{-\frac{1}{2}}$

mathstudent55 · Answer

$\dfrac{d}{dx} x^\color{red}{n} = \color{red}{n} x^{n - 1} $

anonymous · Answer

oh, now it makes sense.

mathstudent55 · Answer

$~~~~~~~~~~~~~~~ = e^\sqrt x \cdot \dfrac{1}{2} x^{-\frac{1}{2}}$

$~~~~~~~~~~~~~~~ =  \dfrac{e^\sqrt x}{2\sqrt x}$

mathstudent55 · Answer

Great!

anonymous · Answer

Thanks for the help!