Differential and Partial differential

Differential of a function

where ${\displaystyle f'(x)}$ is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation

${\displaystyle dy={\frac {dy}{dx}}\,dx}$

holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

${\displaystyle df(x)=f'(x)\,dx.}$

satisfies

${\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon =df(x)+\varepsilon \,}$

where the error ε in the approximation satisfies ε/Δx → 0 as Δx → 0. In other words, one has the approximate identity

${\displaystyle \Delta y\approx dy}$

in which the error can be made as small as desired relative to Δx by constraining Δx to be sufficiently small; that is to say,

${\displaystyle {\frac {\Delta y-dy}{\Delta x}}\to 0}$

as Δx → 0. For this reason, the differential of a function is known as the principal (linear) part in the increment of a function: the differential is a linear function of the increment Δx, and although the error ε may be nonlinear, it tends to zero rapidly as Δx tends to zero.

the partial differential of y with respect to any one of the variables x₁ is the principal part of the change in y resulting from a change dx₁ in that one variable. The partial differential is therefore

${\displaystyle {\frac {\partial y}{\partial x_{1}}}dx_{1}}$

參：

https://en.wikipedia.org/wiki/Differential_of_a_function

全微分和偏微分有什麼不同

假設 z = x^7 + y^3 + 3(x^2)*(y^4) + 4xy + 98 
z為應變數 , x y為自變數 

全微分 : 
就是所有變數通通都要微分 
dz = 7x^6dx + 3y^2dy + 6x(y^4)dx + 12(x^2)*(y^3)dy + 4ydx + 4xdy + 0 
ps. 兩變數相乘的 , 如同一般乘法的微分 , 前微後不微+前不微後微 

偏微分 : 
只對要微的自變數微分 , 其他的都當作常數 
1.對 x 偏微 
∂z/∂x = 7x^6 + 0 + 6x(y^4) + 4y + 0 

2. 對 y 偏微 
∂z/∂y = 0 + 3y^2 + 12(x^2)*(y^3) + 4x + 0

參：

https://tw.answers.yahoo.com/question/index?qid=20050930000013KK04937