See Student[Calculus1] for a general introduction to the Calculus1 subpackage of the Student package.

See SingleStepOverview for an introduction to the step-by-step (or single-step) functionality of the Calculus1 package.

The following table lists the built-in rules for differentiation that do not take parameters.  These rules can be passed as the index to Rule or as a rule argument to Understand.

The name of any univariate function can also be used as a rule argument to the Rule command.  The name of any univariate function recognized by Maple, for example, sin, can be passed as a rule argument to the Understand command (where recognized means that it is of type mathfunc).

There is one differentiation rule which requires a parameter: rewrite.  This rule can be used as the index to a call to Rule, but cannot be given as a rule argument to Understand.  This rule is used to change the form of the expression being differentiated.  It has the general form:

The effect of applying the rewrite rule is to perform each substitution listed as a parameter to the rule, where occurrences of the left-hand side of each substitution are replaced by the corresponding right-hand side.

The main application of this rule is to rewrite an expression of the form ${f\left(x\right)}^{g\left(x\right)}$, where the exponent (at least) depends on the differentiation variable, as an exponential.  The rule would thus be given as:

Note: The Rule routine does not attempt to validate the rewrite rules you provide.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Calculus1}\right)\:$

$\mathrm{infolevel}\left[\mathrm{Student}\left[\mathrm{Calculus1}\right]\right]≔1\:$

$\mathrm{Rule}\left[\mathrm{`*`}\right]\left(\mathrm{Diff}\left(x^2\sin \left(x^2\right)\,x\right)\right)$

Creating problem #1

$\frac{\ⅆ}{\ⅆx}\left(x^2\sin \left(x^2\right)\right)=\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)\sin \left(x^2\right)+x^2\left(\frac{\ⅆ}{\ⅆx}\sin \left(x^2\right)\right)$

$\mathrm{Rule}\left[\mathrm{chain}\right]\left(\right)$

$\frac{\ⅆ}{\ⅆx}\left(x^2\sin \left(x^2\right)\right)=\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)\sin \left(x^2\right)+x^2\left(\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_X0}}\sin \left(\mathrm{\_X0}\right)\right)}{\phantom{\mathrm{\_X0}=x^2}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_X0}}\sin \left(\mathrm{\_X0}\right)\right)}}{\mathrm{\_X0}=x^2}\right)\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)$

$\mathrm{Rule}\left[\sin \right]\left(\right)$

$\frac{\ⅆ}{\ⅆx}\left(x^2\sin \left(x^2\right)\right)=\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)\sin \left(x^2\right)+x^2\cos \left(x^2\right)\left(\frac{\ⅆ}{\ⅆx}\left(x^2\right)\right)$

$\mathrm{Rule}\left[\mathrm{sum}\right]\left(\mathrm{Diff}\left(x^2+\mathrm{Int}\left(\cos \left(t\right)\,t=0..x\right)\,x\right)\right)$

Error, (in Student:-Calculus1:-Rule[sum]) unable to determine which calculus operation is being applied in this problem; you can provide this information as the 2nd argument on your call to Rule or Hint

$\mathrm{Rule}\left[\mathrm{sum}\right]\left(\mathrm{Diff}\left(x^2+\mathrm{Int}\left(\cos \left(t\right)\,t=0..x\right)\,x\right)\,\mathrm{diff}\right)$

Creating problem #2

$\frac{\partial }{\partial x}\left(x^2+{\int }_0^x\cos \left(t\right)\ⅆt\right)=\frac{\ⅆ}{\ⅆx}\left(x^2\right)+\frac{\partial }{\partial x}\left({\int }_0^x\cos \left(t\right)\ⅆt\right)$

$\mathrm{Rule}\left[\mathrm{int}\right]\left(\right)$

$\frac{\partial }{\partial x}\left(x^2+{\int }_0^x\cos \left(t\right)\ⅆt\right)=\frac{\ⅆ}{\ⅆx}\left(x^2\right)+\cos \left(x\right)$

$\mathrm{Rule}\left[\mathrm{`^`}\right]\left(\mathrm{Diff}\left(\exp \left(x\right)\,x\right)\right)$

Creating problem #3

Rule [power] does not apply

$\frac{\ⅆ}{\ⅆx}{\ⅇ}^x=\frac{\ⅆ}{\ⅆx}{\ⅇ}^x$

$\mathrm{Rule}\left[\mathrm{rewrite},x^{\sin \left(x\right)}=\exp \left(\sin \left(x\right)\ln \left(x\right)\right)\right]\left(\mathrm{Diff}\left(x^{\sin \left(x\right)}\,x\right)\right)$

Creating problem #4

$\frac{\ⅆ}{\ⅆx}\left(x^{\sin \left(x\right)}\right)=\frac{\ⅆ}{\ⅆx}{\ⅇ}^{\sin \left(x\right)\ln \left(x\right)}$

$\mathrm{Rule}\left[\mathrm{`*`}\right]\left(\mathrm{Diff}\left(rf\left(r\right)\,r\right)\right)$

Creating problem #5

$\frac{\ⅆ}{\ⅆr}\left(rf\left(r\right)\right)=\left(\frac{\ⅆr}{\ⅆr}\right)f\left(r\right)+r\left(\frac{\ⅆ}{\ⅆr}f\left(r\right)\right)$

$\mathrm{Rule}\left[f\right]\left(\right)$

$\frac{\ⅆ}{\ⅆr}\left(rf\left(r\right)\right)=\left(\frac{\ⅆr}{\ⅆr}\right)f\left(r\right)+r\left(\frac{\ⅆ}{\ⅆr}f\left(r\right)\right)$

$\mathrm{Rule}\left[\mathrm{identity}\right]\left(\right)$

$\frac{\ⅆ}{\ⅆr}\left(rf\left(r\right)\right)=f\left(r\right)+r\left(\frac{\ⅆ}{\ⅆr}f\left(r\right)\right)$

$\mathrm{ShowIncomplete}\left(\right)$

The current problem is complete