Derivative of e^(x/2)

The calculator will find the derivative of

e^{\frac{x}{2}}

, with steps shown.

Your Input

Find $\frac{d}{dx} \left(e^{\frac{x}{2}}\right)$ .

Solution

The function $e^{\frac{x}{2}}$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = e^{u}$ and $g{\left(x \right)} = \frac{x}{2}$ .

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$ :

{\color{red}\left(\frac{d}{dx} \left(e^{\frac{x}{2}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\frac{x}{2}\right)\right)}

The derivative of the exponential is $\frac{d}{du} \left(e^{u}\right) = e^{u}$ :

{\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(\frac{x}{2}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(\frac{x}{2}\right)

Return to the old variable:

e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\frac{x}{2}\right) = e^{{\color{red}\left(\frac{x}{2}\right)}} \frac{d}{dx} \left(\frac{x}{2}\right)

Apply the constant multiple rule $\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$ with $c = \frac{1}{2}$ and $f{\left(x \right)} = x$ :

e^{\frac{x}{2}} {\color{red}\left(\frac{d}{dx} \left(\frac{x}{2}\right)\right)} = e^{\frac{x}{2}} {\color{red}\left(\frac{\frac{d}{dx} \left(x\right)}{2}\right)}

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 1$ , in other words, $\frac{d}{dx} \left(x\right) = 1$ :

\frac{e^{\frac{x}{2}} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{2} = \frac{e^{\frac{x}{2}} {\color{red}\left(1\right)}}{2}

Thus, $\frac{d}{dx} \left(e^{\frac{x}{2}}\right) = \frac{e^{\frac{x}{2}}}{2}$ .

Answer

$\frac{d}{dx} \left(e^{\frac{x}{2}}\right) = \frac{e^{\frac{x}{2}}}{2}$ A

Derivative of ex2e^{\frac{x}{2}}e2x​

Your Input

Solution

Answer

Derivative of $e^{\frac{x}{2}}$