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real_function_derivative [2016/04/13 09:26] nikolaj |
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| 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: | ||
| + | |||
| + | ^ $\frac{\partial f(x)}{\partial x}(g(x)) \equiv\frac{\partial f}{\partial x}(g(x))\equiv\frac{\partial f(x)}{\partial g(x)} \equiv\frac{\partial f}{\partial g(x)} \equiv \frac{\partial}{\partial y}|_{y=g(x)}\ f(y) \equiv\left(\frac{\partial}{\partial y}f(y)\right)_{y=g(x)}$ ^ | ||
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| + | 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 === | ||
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| 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 === | ||
