Notation

Scientific Notebook recognizes various notations for the derivative, including the forms

$\displaystyle {\frac{{d}}{{dx}}}$,  $\displaystyle {\frac{{d^{n}}}{{dx^{n}}}}$,  Dx,  Dxx,  Dx2,  Dxy,  Dxsyt,  $\displaystyle {\frac{{%
\partial }}{{\partial x}}}$, and $\displaystyle {\frac{{\partial ^{n}}}{{\partial
x^{s}\partial y^{t}}}}$

where s + t = n. If f (x) is defined as a function, then the forms f(x), f′′(x), and f(n)(x) are also recognized.

$\blacktriangleright$ To find the derivative of x2

You get the same result from any of the following expressions.

    $\displaystyle {\frac{{dx^2}}{{dx}}}$        $\displaystyle {\frac{d}{{dx}}}$x2        $\displaystyle {\frac{d}{{dx}}}$$\displaystyle \left(\vphantom{ x^2}\right.$x2$\displaystyle \left.\vphantom{ x^2}\right)$        $\displaystyle {\frac{d}{{dx}}}$$\displaystyle \left[\vphantom{ x^2}\right.$x2$\displaystyle \left.\vphantom{ x^2}\right]$        $\displaystyle {\frac{d}{{dx}}}$$\displaystyle \left\{\vphantom{ x^2}\right.$x2$\displaystyle \left.\vphantom{ x^2}\right\}$  
    Dxx2        Dx$\displaystyle \left(\vphantom{ x^2}\right.$x2$\displaystyle \left.\vphantom{ x^2}\right)$        $\displaystyle {\frac{{\partial x^2}}{{\partial x}%
}}$        $\displaystyle {\frac{\partial}{{\partial x}}}$x2        $\displaystyle {\frac{\partial}{{\partial x}}}$$\displaystyle \left(\vphantom{ x^2}\right.$x2$\displaystyle \left.\vphantom{ x^2}\right)$  

Note that the ``prime'' notation works only for defined functions. For example, Evaluate applied to $\left(\vphantom{ x+\sin x}\right.$x + sin x$\left.\vphantom{ x+\sin x}\right)^{{\prime }}_{}$ does not give the derivative.

$\blacktriangleright$ Evaluate

$\left(\vphantom{ x+\sin x}\right.$x + sin x$\left.\vphantom{ x+\sin x}\right)^{{\prime }}_{}$ = x + sin x

The derivative is applied to the term directly to the right of the operator, as illustrated in the following two examples.

$\blacktriangleright$ Evaluate

${\dfrac{{\partial ^{2}}}{{\partial x^{2}}}}$x2 +3x = 2 + 3x 6pt

${\dfrac{{\partial ^{2}}}{{\partial x^{2}}}}$$\left(\vphantom{ x^{2}+3x}\right.$x2 + 3x$\left.\vphantom{ x^{2}+3x}\right)$ = 2

Good notation is important when you are using Scientific Notebook, just as good notation is important in mathematics. Ambiguous notation may be accepted, but it may lead to an unexpected output. Experiment with expressions such as

$\displaystyle {\frac{{\partial ^{2}}}{{\partial x^{2}}}}$$\displaystyle \left(\vphantom{ (x^{2}+3x}\right.$(x2 + 3x$\displaystyle \left.\vphantom{ (x^{2}+3x}\right)$         and        $\displaystyle {\frac{{\partial ^{2}}}{{\partial x^{2}}}}${x2 + 3x

to see how Scientific Notebook interprets ill-formed expressions.

Tip    Making good use of the parentheses button itbpF0.3009in0.3009in0.0701inparens.wmf eliminates many common types of ill-formed expressions.

You can find derivatives of piecewise-defined functions. The result will be a piecewise-defined function.


\begin{example}
Make the definition
\begin{displaymath}
f(x)=\left\{
\begi...
...
6x & \func{if} & 0<x
\end{array}
\right.
\end{displaymath}
\end{example}


\begin{example}
It is not necessary to name a piecewise function in order to ta...
...^{2}} & \func{if} & 1<x
\end{array}
\right.
\end{displaymath}
\end{example}