Generic Functions

It is possible to allow Scientific Notebook to treat f (x) as an arbitrary, or ``generic'' function. Simplify define f (x) to be a function, without associating it with a formula.

$\blacktriangleright$ Define + New Definition, or click itbpF0.3001in0.3001in0.0701innewdef.wmf

f (x)

g(x)

Standard rules of calculus apply to arbitrary functions.

$\blacktriangleright$ Evaluate

${\dfrac{{d}}{{dx}}}$$\left(\vphantom{ f\left( g(x)\right) }\right.$f$\left(\vphantom{ g(x)}\right.$g(x)$\left.\vphantom{ g(x)}\right)$$\left.\vphantom{ f\left( g(x)\right) }\right)$ = f′$\left(\vphantom{ g\left( x\right) }\right.$g$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$$\left.\vphantom{ g\left( x\right) }\right)$g′$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ 6pt

Dx$\left[\vphantom{ f(x)g(x)}\right.$f (x)g(x)$\left.\vphantom{ f(x)g(x)}\right]$ = $\left[\vphantom{ f^{\prime }\left( x\right) g\left( x\right) +f\left( x\right) g^{\prime }\left( x\right) }\right.$f′$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$g$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ + f$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$g′$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$$\left.\vphantom{ f^{\prime }\left( x\right) g\left( x\right) +f\left( x\right) g^{\prime }\left( x\right) }\right]$ 6pt

Dx${\dfrac{{f(x)}}{{g(x)}}}$ = ${\dfrac{{f^{\prime }\left( x\right) }}{{% g\left( x\right) }}}$ - ${\dfrac{{f\left( x\right) }}{{g^{2}\left( x\right) }}}$g′$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ 6pt

${\dfrac{{d}}{{dx}}}$$\dint_{{0}}^{{x}}$f (t) dt = f$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$

$\blacktriangleright$ Series, Expand in Powers of: x

f (x) = f$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ + f′$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$x + ${\frac{{1}}{{2}% }}$f′′$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$x2 + ${\frac{{1}}{{6}}}$f′′′$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$x3 + O$\left(\vphantom{ x^{4}}\right.$x4$\left.\vphantom{ x^{4}}\right)$