導數計算器
逐步計算導數
此線上計算器會依據常見微分法則（乘積法則、商法則、鏈式法則等）計算任意函數的導數，並顯示步驟。它可處理多項式、有理、無理、指數、對數、三角、反三角、雙曲與反雙曲函數。亦可在需要時於給定點處評估導數。並支援計算一階、二階、三階導數，最高可至第 10 階。
相關計算器: 對數微分計算器, 隱式微分計算器（附步驟）
您的輸入
求$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right)$$$。
解答
將乘積法則 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ 應用於 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = \sin{\left(2 x \right)}$$$：
$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(2 x \right)} + x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right) + \sin{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(\sin{\left(2 x \right)}\right) + \sin{\left(2 x \right)} {\color{red}\left(1\right)}$$函數 $$$\sin{\left(2 x \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = 2 x$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} + \sin{\left(2 x \right)} = x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)}$$正弦函數的導數為$$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$：
$$x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} = x {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)}$$返回原變數：
$$x \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)} = x \cos{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right) + \sin{\left(2 x \right)}$$套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = 2$$$ 與 $$$f{\left(x \right)} = x$$$：
$$x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} + \sin{\left(2 x \right)} = x \cos{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$2 x \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \sin{\left(2 x \right)} = 2 x \cos{\left(2 x \right)} {\color{red}\left(1\right)} + \sin{\left(2 x \right)}$$因此，$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right) = 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}$$$。
答案
$$$\frac{d}{dx} \left(x \sin{\left(2 x \right)}\right) = 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}$$$A