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Step-by-Step Solutions


Simplification Steps

Steps for Sketching a Curve

Simplification Steps

Maple 2022 includes new commands for showing the steps needed to manipulate algebraic expressions in order to reduce them to their simplest form.

In general the generated steps try to find that hard balance between being too verbose and too cryptic.  The SimplifySteps command errs on the side of adding more steps, and is aimed to help someone who wants to learn what the steps are, even for fundamentals like adding fractions.  Depending on the problem it will adjust somewhat; recognizing higher level problems for which it will decide to skip more easy-level steps.

Here are some examples of different categories of problems that SimplifySteps as well as related command FractionSteps can handle:





There is a dedicated FractionSteps command that goes into slightly more detail than SimplifySteps

FractionSteps 1/2 + 1/6

Let's Simplify FractionsFind fractions to get lowest common denominator of6MultiplyAdd numeratorsMultiply31Multiply11Add3+1Cancel out factor of2



SimplifySteps 1/2 + 1/6

Let's simplifyFind fractions to get lowest common denominator of6MultiplyAdd fractions23




SimplifySteps sqrt(6)*sqrt(10)/sqrt(12) 

Let's simplifyPull out a factor of4=2from12Multiply in order to rationalize the denominatorMultiply the denominatorFactor rootsCombine22,33Multiply325Cancel out factor of65



Let's simplifyMultiply in order to rationalize the denominatorMultiply the denominatorFactor rootsCombine3233,21322Multiply331622135Multiply265213316Cancel out factor of125213316





Let's simplifyEvaluate exponent3xy22Multiply2x3y39x2y418x5y7



Let's simplifyApply the product ruleanam=an+mto add exponents with common baseAdd exponentsSolution



Let's simplifyApply the integer power of a power rule,anm=anm



Let's simplifyCancel out factor ofy4providedy40y



Let's simplifyDivide assumingy4≠ 01y9



Let's simplifyApply the quotient rule:anam=anmEvaluate exponent1231123




SimplifySteps log[10](2)/log[10](5)+log[10](3) 

Let's simplifyUse the log rule,logax=logbxlogbato express as a single logarithmSolution


Note: This is different than how Maple's simplify command treats expressions like this, always converting to ln:






SimplifySteps %log10100+%log101000 

Let's simplifyEvaluatelog10100Evaluatelog101000Add2+35



Let's simplifyApply the log rulelogamn=nlogamApply the log rulelogaa=1Multiply5420



You can also use the new PowerSteps command to get step by step results for problems with radicals, exponents, and logarithms.  


SimplifySteps 1+cot(x)^2 

Let's simplifyApplyPythagorastrig identity,cotx2=cscx211+cscx21ApplyReciprocal Functiontrig identity,cscx=1sinx1sinx2Evaluate1sinx2


SimplifySteps"sinPi + Pi + x"

Let's simplifyAddπ+π+xEvaluatesin2π+xsinx



Let's simplifyApplyReciprocal Functiontrig identity,secx=1cosxsecx211cosx2ApplyPythagorastrig identity,secx2=1+tanx21+tanx21cosx2ApplyQuotienttrig identity,tanx=sinxcosxsinxcosx2cosx2Evaluatesinx2



Let's simplifyApplyPythagorastrig identity,cotx2=cscx21cosx2+11+cscx21ApplyReciprocal Functiontrig identity,cscx=1sinxcosx2+11sinx2ApplyPythagorastrig identity,cosx2=1sinx21sinx2+1sinx2Evaluate1



You can also use the new TrigSteps command to get these step by step results.  


Showing the steps to solving an integral, limit, or derivative has been available in past versions of Maple via the Student:-Calculus1:-ShowSolution command. You can now also access those step by step solutions through SimplifySteps, further unifying the ability to do step by step solutions using a single command.


Let's simplifyIntegralto evaluatexⅆx1. Apply thepowerrule to the termxⅆxRecall the definition of thepowerrule, for n-1xⅆx=This means:xⅆx=So,xⅆx=x22We can rewrite the integral as:x22Sub evaluatedintegralback in expression3x22



Steps for Sketching a Curve

Maple 2022 includes a new command for showing the steps needed to sketch the graph of an expression by identifying the basic function and the transformations done to the function.  Various kinds of expressions are handled, including trig, logs, and polynomials to pick just a few.  Here are some examples:



Let's plot2sin3x+π3+1Compared to the plot ofsinx, we have a vertical stretch by a factor of2Then, we have a horizontal compression by a factor of13Then, we have a vertical shift of1Then, we have a horizontal shift ofπ9Apply the horizontal shift and stretch to the range,x=2π..2π+π9..+π9=−2.443460953..1.745329252We can now plot using the information extractedPLOT...



Let's plot2x2+4x+10Complete the square2x+12+8With the expression in vertex form we can extract valuable informationThe coefficient2of thex+12term indicates a parabola that opensupand has a verticalstretchof2We have a horizontal shift of−1and a vertical shift of8which gives a vertex of (−1,8)We can now plot using the information extractedPLOT...



Let's plot4x+10This is a line; find two points and draw a line through themy=4x+10Setx= 0 to solve for y intercepty=10This gives a y intercept of (0,10)y=10Set expresson to 0 to solve forxintercept0=Subtract4xfrom both sides=Simplify=10Divide both sides by−4=Simplifyx=52This gives anxintercept of (52,0)x=52By connecting through the two points we can plot the linePLOT...



Let's plot23x+2Rewrite the equation in the following formCompared to the plot of1x, we have a vertical stretch by a factor of2PLOT...Then, we have a horizontal compression by a factor of13PLOT...Then, we have a horizontal shift of23PLOT...The final plot with asymptotes in cyan aty=0andx=23isPLOT...



For more information, see the help page Student:-Basics:-CurveSketchSteps.