Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
real_function_derivative [2016/04/13 09:26]
nikolaj
real_function_derivative [2016/05/31 20:42] (current)
nikolaj
Line 12: Line 12:
  
 In this entry we defined how to pass from a given function $f$ to the derivative $f'$. This process is in fact functional in $f$ and can hence be internalized,​ see [[Fréchet derivative]]. In this entry we defined how to pass from a given function $f$ to the derivative $f'$. This process is in fact functional in $f$ and can hence be internalized,​ see [[Fréchet derivative]].
 +
 +=== Notation ===
 +^ $f'(x) \equiv \frac{\partial f(x)}{\partial x} \equiv \frac{\partial f}{\partial x} $ ^
 +
 +More generally we can write the differential with successive evaluation of the derivative at a point $g(x)$ in the following ways:
 +
 +^ $\frac{\partial f(x)}{\partial x}(g(x)) \equiv\frac{\partial f}{\partial x}(g(x))\equiv\frac{\partial f(x)}{\partial g(x)} \equiv\frac{\partial f}{\partial g(x)} \equiv \frac{\partial}{\partial y}|_{y=g(x)}\ f(y) \equiv\left(\frac{\partial}{\partial y}f(y)\right)_{y=g(x)}$ ^
 +
 +Note that $\frac{\partial f(x)}{\partial g(x)}$ or $\frac{\partial f}{\partial g(x)}$ might easily be misread: ​
 +The expression $\frac{\partial \mathrm{exp}(x^6)}{\partial x^2}$ is taken to be $\left(\frac{\partial}{\partial y}\mathrm{exp}(y^6)\right)_{y=x^2}$ and not $\left(\frac{\partial}{\partial y}\mathrm{exp}(x^3)\right)_{y=x^2}$. I.e. functions are always derived w.r.t. their proper arguments alone.
 +
 +If the domain of $f$ is higher dimensional,​ some of the above notations don't work anymore. We must e.g. use the unambigous notation $\frac{\partial f(x^1,​x^2)}{\partial x^2}(g_1(x^1,​x^2),​g_2(x^1,​x^2))$. Since this is cumbersome, the variable names are usually implicitly understood to be held fixed, e.g. in writing "​$f(x,​y)$"​ once and then have $\frac{\partial f}{\partial y}$, denote $\frac{\partial f(x,​y)}{\partial y}(x,y)$.
  
 === Theorems === === Theorems ===
Line 45: Line 57:
 d[#^n &][x] d[#^n &][x]
 </​code>​ </​code>​
- 
-=== Notation === 
-^ $f'(x) \equiv \frac{\partial f(x)}{\partial x} \equiv \frac{\partial f}{\partial x} $ ^ 
- 
-More generally we can write the differential with successive evaluation of the derivative at a point $g(x)$ in the following ways: 
- 
-^ $\frac{\partial f(x)}{\partial x}(g(x)) \equiv\frac{\partial f}{\partial x}(g(x))\equiv\frac{\partial f(x)}{\partial g(x)} \equiv\frac{\partial f}{\partial g(x)} \equiv \frac{\partial}{\partial y}|_{y=g(x)}\ f(y) \equiv\left(\frac{\partial}{\partial y}f(y)\right)_{y=g(x)}$ ^ 
- 
-Note that $\frac{\partial f(x)}{\partial g(x)}$ or $\frac{\partial f}{\partial g(x)}$ might easily be misread: ​ 
-The expression $\frac{\partial \mathrm{exp}(x^6)}{\partial x^2}$ is taken to be $\left(\frac{\partial}{\partial y}\mathrm{exp}(y^6)\right)_{y=x^2}$ and not $\left(\frac{\partial}{\partial y}\mathrm{exp}(x^3)\right)_{y=x^2}$. I.e. functions are always derived w.r.t. their proper arguments alone. 
- 
-If the domain of $f$ is higher dimensional,​ some of the above notations don't work anymore. We must e.g. use the unambigous notation $\frac{\partial f(x^1,​x^2)}{\partial x^2}(g_1(x^1,​x^2),​g_2(x^1,​x^2))$. Since this is cumbersome, the variable names are usually implicitly understood to be held fixed, e.g. in writing "​$f(x,​y)$"​ once and then have $\frac{\partial f}{\partial y}$, denote $\frac{\partial f(x,​y)}{\partial y}(x,y)$. 
  
 === Reference === === Reference ===
Link to graph
Log In
Improvements of the human condition