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Derivative, partial derivative, directional derivative
2022-07-07 02:37:00 【Allenpandas】
1. Derivative and partial derivative
1.1. derivative
Derivative defines : The reaction is a function y = f ( x ) y=f(x) y=f(x) At some point along the argument x x x In the right direction ( namely : x x x Affirmative direction ) The rate of change of .
Derivative formula :
function y = f ( x ) y=f(x) y=f(x) stay x 0 x_0 x0 The derivative of a point is written as f ′ ( x 0 ) f'(x_0) f′(x0), be f ′ ( x 0 ) f'(x_0) f′(x0) by :
f ′ ( x 0 ) = lim Δ x → 0 Δ y Δ x = lim Δ x → 0 f ( x 0 + Δ x ) − f ( x 0 ) Δ x f' (x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} f′(x0)=Δx→0limΔxΔy=Δx→0limΔxf(x0+Δx)−f(x0)
Geometric meaning : function y = f ( x ) y=f(x) y=f(x) stay x 0 x_0 x0 The derivative of a point f ′ ( x 0 ) f'(x_0) f′(x0) It means that the function curve is at the point P 0 ( x 0 , f ( x 0 ) ) P_0(x_0, f(x_0)) P0(x0,f(x0)) The slope of the tangent at 【 The geometric meaning of the derivative is that the function curve is at this point P 0 ( x 0 , f ( x 0 ) ) P_0(x_0, f(x_0)) P0(x0,f(x0)) The tangent slope on the 】.
1.2. Partial derivative
Partial derivative definition : Take the binary function for example , The reaction is a function z = f ( x , y ) z=f(x,y) z=f(x,y) At a certain point along a positive axis ( namely : Along x x x The axis is in the positive direction or along y y y Affirmative direction ) The rate of change of .
Partial derivative formula :
In binary function z = f ( x , y ) z=f(x, y) z=f(x,y) For example :
function z = f ( x , y ) z=f(x,y) z=f(x,y) stay ( x 0 , y 0 ) (x_0, y_0) (x0,y0) Point to point pair x x x The partial derivative of is written as ∂ z ∂ x \frac{\partial z}{\partial x} ∂x∂z( It can also be recorded as : ∂ f ∂ x \frac{\partial f}{\partial x} ∂x∂f , z x z_x zx or f x ( x , y ) f_x(x,y) fx(x,y)), be ∂ z ∂ x \frac{\partial z}{\partial x} ∂x∂z by :
∂ z ∂ x = lim Δ x → 0 f ( x 0 + Δ x , y 0 ) − f ( x 0 , y 0 ) Δ x \frac{\partial z}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0) - f(x_0, y_0)}{\Delta x} ∂x∂z=Δx→0limΔxf(x0+Δx,y0)−f(x0,y0)
function z = f ( x , y ) z=f(x,y) z=f(x,y) stay ( x 0 , y 0 ) (x_0, y_0) (x0,y0) Point to point pair y y y The partial derivative of is written as ∂ z ∂ y \frac{\partial z}{\partial y} ∂y∂z( It can also be recorded as : ∂ f ∂ y \frac{\partial f}{\partial y} ∂y∂f , z y z_y zy or f y ( x , y ) f_y(x,y) fy(x,y)), be ∂ z ∂ y \frac{\partial z}{\partial y} ∂y∂z by :
∂ z ∂ y = lim Δ y → 0 f ( x 0 , y 0 + Δ y ) − f ( x 0 , y 0 ) Δ y \frac{\partial z}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x_0, y_0 + \Delta y) - f(x_0, y_0)}{\Delta y} ∂y∂z=Δy→0limΔyf(x0,y0+Δy)−f(x0,y0)
notes : The essence of derivative and partial derivative is ⼀ To , All when ⾃ The change of variables tends to 0 when , The change of function value is related to ⾃ Changes in variables , Between them ⽐ Limit of value .
3. Directional derivative
before ⾯ derivative and Partial derivative The definition of , All along The coordinate axis is positive ⽅ towards Discuss the rate of change of the function . So when we talk about functions Arbitrarily along ⽅ Rate of change in direction when , That leads to ⽅ Definition of derivative .
Directional derivative : The reaction is a function y y y At some point x 0 x_0 x0 Along a specific direction ( Is not necessarily x x x The axis is in the right direction ) The rate of change of .
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