f ( x ) = ∣ x ∣ = { − x , x < 0 x , x ≥ 0 = { − x , x < 0 0 , x = 0 − x , x < 0 f(x)=\left|x\right| = \left\{ \begin{aligned} -x, x < 0 \\ x, x \geq 0 \end{aligned} \right. = \left\{ \begin{aligned} -x, x < 0 \\ 0, x = 0 \\ -x, x < 0 \end{aligned} \right. f(x)=∣x∣={−x,x x_0 \end{aligned}\right. f(x)={f(x),x≤x0​g(x),x>x0​​

或

f ( x ) = { f ( x ) , x < x 0 a , x = x 0 g ( x ) , x > x 0 f(x) = \left \{ \begin{aligned} f(x), x < x_0 \\ a, x = x_0 \\ g(x), x > x_0\end{aligned} \right. f(x)=⎩⎪⎨⎪⎧​f(x),xx0​​

D ( x ) = { 1 , x ∈ Q 0 , x ∈ Q ∁ , Q 表 示 有 理 数 , Q ∁ 表 示 有 理 数 的 补 集 D(x) = \left \{ \begin{aligned} 1 , x \in \mathbb{Q} \\ 0 , x \in \mathbb{Q}^\complement \end{aligned} \right. ,\qquad \mathbb{Q}表示有理数, \mathbb{Q}^\complement表示有理数的补集 D(x)={1,x∈Q0,x∈Q∁​,Q表示有理数,Q∁表示有理数的补集

f ( x ) = u ( x ) v ( x ) , u ( x ) > 0 , 且 u ( x ) ≠ 1 f(x) = u(x)^{v(x)}, u(x) > 0, 且u(x) \neq 1 f(x)=u(x)v(x),u(x)>0,且u(x)​=1

可以推出：

u ( x ) v ( x ) = e v ( x ) ⋅ ln ⁡ u ( x ) u(x)^{v(x)} = e^{v(x) \cdot \ln u(x)} u(x)v(x)=ev(x)⋅lnu(x)

用于幂指函数求导。