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Derivative of $$$e^{\frac{x}{3}}$$$
Related calculator: Derivative Calculator
Solution
The function $$$e^{\frac{x}{3}}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = e^{u}$$$ and $$$g{\left(x \right)} = \frac{x}{3}$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{\frac{x}{3}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\frac{x}{3}\right)\right)}$$The derivative of the exponential is $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(\frac{x}{3}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(\frac{x}{3}\right)$$Return to the old variable:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\frac{x}{3}\right) = e^{{\color{red}\left(\frac{x}{3}\right)}} \frac{d}{dx} \left(\frac{x}{3}\right)$$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 $$$c = \frac{1}{3}$$$ and $$$f{\left(x \right)} = x$$$:
$$e^{\frac{x}{3}} {\color{red}\left(\frac{d}{dx} \left(\frac{x}{3}\right)\right)} = e^{\frac{x}{3}} {\color{red}\left(\frac{\frac{d}{dx} \left(x\right)}{3}\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{e^{\frac{x}{3}} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{3} = \frac{e^{\frac{x}{3}} {\color{red}\left(1\right)}}{3}$$Thus, $$$\frac{d}{dx} \left(e^{\frac{x}{3}}\right) = \frac{e^{\frac{x}{3}}}{3}$$$.
Answer
$$$\frac{d}{dx} \left(e^{\frac{x}{3}}\right) = \frac{e^{\frac{x}{3}}}{3}$$$A