Turunan dari $$$e^{2 x} + 1$$$

Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:

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

Fungsi $$$e^{2 x}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = e^{u}$$$ dan $$$g{\left(x \right)} = 2 x$$$.

Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

Turunan dari fungsi eksponensial adalah $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:

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

Kembalikan ke variabel semula:

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

Turunan dari suatu konstanta adalah $$$0$$$:

$$e^{2 x} \frac{d}{dx} \left(2 x\right) + {\color{red}\left(\frac{d}{dx} \left(1\right)\right)} = e^{2 x} \frac{d}{dx} \left(2 x\right) + {\color{red}\left(0\right)}$$

Terapkan aturan kelipatan konstanta $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ dengan $$$c = 2$$$ dan $$$f{\left(x \right)} = x$$$:

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

Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

Dengan demikian, $$$\frac{d}{dx} \left(e^{2 x} + 1\right) = 2 e^{2 x}$$$.