Derivative of $$$e^{2 t}$$$

The calculator will find the derivative of $$$e^{2 t}$$$, with steps shown.

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Find $$$\frac{d}{dt} \left(e^{2 t}\right)$$$.


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

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

$${\color{red}\left(\frac{d}{dt} \left(e^{2 t}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dt} \left(2 t\right)\right)}$$

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

$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dt} \left(2 t\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dt} \left(2 t\right)$$

Return to the old variable:

$$e^{{\color{red}\left(u\right)}} \frac{d}{dt} \left(2 t\right) = e^{{\color{red}\left(2 t\right)}} \frac{d}{dt} \left(2 t\right)$$

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

$$e^{2 t} {\color{red}\left(\frac{d}{dt} \left(2 t\right)\right)} = e^{2 t} {\color{red}\left(2 \frac{d}{dt} \left(t\right)\right)}$$

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

$$2 e^{2 t} {\color{red}\left(\frac{d}{dt} \left(t\right)\right)} = 2 e^{2 t} {\color{red}\left(1\right)}$$

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


$$$\frac{d}{dt} \left(e^{2 t}\right) = 2 e^{2 t}$$$A