Turunan dari $$$a^{\sqrt{x}}$$$ terhadap $$$x$$$
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Masukan Anda
Temukan $$$\frac{d}{dx} \left(a^{\sqrt{x}}\right)$$$.
Solusi
Fungsi $$$a^{\sqrt{x}}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = a^{u}$$$ dan $$$g{\left(x \right)} = \sqrt{x}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(a^{\sqrt{x}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(a^{u}\right) \frac{d}{dx} \left(\sqrt{x}\right)\right)}$$Terapkan aturan eksponen $$$\frac{d}{du} \left(n^{u}\right) = n^{u} \ln\left(n\right)$$$ dengan $$$n = a$$$:
$${\color{red}\left(\frac{d}{du} \left(a^{u}\right)\right)} \frac{d}{dx} \left(\sqrt{x}\right) = {\color{red}\left(a^{u} \ln\left(a\right)\right)} \frac{d}{dx} \left(\sqrt{x}\right)$$Kembalikan ke variabel semula:
$$a^{{\color{red}\left(u\right)}} \ln\left(a\right) \frac{d}{dx} \left(\sqrt{x}\right) = a^{{\color{red}\left(\sqrt{x}\right)}} \ln\left(a\right) \frac{d}{dx} \left(\sqrt{x}\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = \frac{1}{2}$$$:
$$a^{\sqrt{x}} \ln\left(a\right) {\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)} = a^{\sqrt{x}} \ln\left(a\right) {\color{red}\left(\frac{1}{2 \sqrt{x}}\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(a^{\sqrt{x}}\right) = \frac{a^{\sqrt{x}} \ln\left(a\right)}{2 \sqrt{x}}$$$.
Jawaban
$$$\frac{d}{dx} \left(a^{\sqrt{x}}\right) = \frac{a^{\sqrt{x}} \ln\left(a\right)}{2 \sqrt{x}}$$$A