单变量导数的公式如下：
f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x ( 1 ) f^{'}(x)=\lim \limits_{ \Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} \qquad (1) f′(x)=Δx→0limΔxf(x+Δx)−f(x)(1)
（1）式等价于
f ′ ( x ) ≈ f ( x + Δ x ) − f ( x ) Δ x Δ x 取 很 小 的 值 f^{'}(x) \approx \frac{f(x+\Delta x)-f(x)}{\Delta x} \qquad \Delta x取很小的值 f′(x)≈Δxf(x+Δx)−f(x)Δx取很小的值
即 f ( x + Δ x ) − f ( x ) ≈ f ′ ( x ) ⋅ Δ x f(x+\Delta x)-f(x) \approx f^{'}(x) ·\Delta x f(x+Δx)−f(x)≈f′(x)⋅Δx
以此推广到多变量函数的近似公式
f ( x + Δ x , y + Δ y ) − f ( x , y ) ≈ ∂ f ∂ x ⋅ Δ x + ∂ f ∂ y ⋅ Δ y f(x+\Delta x,y+\Delta y)-f(x,y) \approx \frac{\partial f}{\partial x} ·\Delta x+\frac{\partial f}{\partial y} ·\Delta y f(x+Δx,y+Δy)−f(x,y)≈∂x∂f⋅Δx+∂y∂f⋅Δy
令 Δ z = f ( x + Δ x , y + Δ y ) − f ( x , y ) \Delta z=f(x+\Delta x,y+\Delta y)-f(x,y) Δz=f(x+Δx,y+Δy)−f(x,y)，即 Δ z = ∂ f ∂ x ⋅ Δ x + ∂ f ∂ y ⋅ Δ y \Delta z=\frac{\partial f}{\partial x} ·\Delta x+\frac{\partial f}{\partial y} ·\Delta y Δz=∂x∂f⋅Δx+∂y∂f⋅Δy
表示成内积的形式即为：
Δ z = ( ∂ f ∂ x , ∂ f ∂ y ) . ( Δ x , Δ y ) \Delta z=(\frac{\partial f}{\partial x} ,\frac{\partial f}{\partial y} ).(\Delta x,\Delta y) Δz=(∂x∂f,∂y∂f).(Δx,Δy)