randRangeNonZero(-2,2) randRangeNonZero(-4,4) (A>0?-1:1)*randRange(3,8) randFromArray(["\\Delta x","h"])
randRange(-6,6)

מהו השיפוע של הישר המשיק ל- f(x) = expr(["+",["*",A,["^","x",2]],["*",B,"x"],C]) בנקודה x = X?

init({range:[[-10,10],[-10,10]],scale:[20,20]}),grid([-10,10],[-10,10],{stroke:"#e5e5e5"}),style({stroke:"#888",strokeWidth:2,arrows:"->"},function(){line([-10,0],[10,0]),line([0,-10],[0,10])}),plot(function(e){return(A*e+B)*e+C},[-10,10],{stroke:"#6495ED"}),circle([X,(A*X+B)*X+C],.15,{fill:"black",stroke:"none"})
2*A*X+B
plot(function(e){return(2*A*X+B)*(e-X)+(A*X+B)*X+C},[-10,10],{stroke:"black",strokeWidth:1})

השיפוע של הישר המשיק הוא \displaystyle \lim_{H \to 0} \frac{f(x + H) - f(x)}{H}.

\qquad = \displaystyle \lim_{H \to 0} \frac{(expr(["+",["*",A,["^",["+","x",H],2]],["*",B,["+","x",H]],C])) - (expr(["+",["*",A,["^","x",2]],["*",B,"x"],C]))}{H}

\qquad = \displaystyle \lim_{H \to 0} \frac{(expr(["+",["*",A,["+",["^","x",2],"2x "+H,["^",H,2]]],["*",B,["+","x",H]],C])) - (expr(["+",["*",A,["^","x",2]],["*",B,"x"],C]))}{H}

\qquad = \displaystyle \lim_{H \to 0} \frac{expr(["+",["*",A,["^","x",2]],["*",2*A,"x "+H],["*",A,["^",H,2]],["*",B,"x"],["*",B,H],C,["*",-A,["^","x",2]],["*",-B,"x"],-C])}{H}

\qquad = \displaystyle \lim_{H \to 0} \frac{expr(["+",["*",2*A,"x "+H],["*",A,["^",H,2]],["*",B,H]])}{H}

\qquad = \displaystyle \lim_{H \to 0} expr(["+",["*",2*A,"x"],["*",A,H],B])

\qquad = \displaystyle expr(["+",["*",2*A,"x"],B])

\qquad = \displaystyle expr(["+",["*",2*A,X],B])

\qquad = 2*A*X+B

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