Übung: Quotientenregel
generateSpecialFunction("x")
generateSpecialFunction("x")
FUNCN.fText
FUNCN.ddxFText
FUNCD.fText
FUNCD.ddxFText
funcNotation("x")
function(a, b, c, d, e, min) {
var term1 = "\\left(" + a + "\\right)" +
(a === b ? "^2" : "\\left(" + b + "\\right)");
var term2 = "\\left(" + c + "\\right)" +
(c === d ? "^2" : "\\left(" + d + "\\right)");
return "\\dfrac{" + term1 + min + term2 + "}" +
"{\\left(" + e + "\\right)^2}";
}
Bestimme \displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \frac{FUNCN.fText}{FUNCD.fText} \biggr).
ANSWER( N_DF, D_F, D_DF, N_F, D_F, "-" )
ANSWER( N_DF, D_DF, D_F, N_F, D_F, "-" )ANSWER( N_DF, D_F, D_DF, N_F, N_F, "-" )ANSWER( N_DF, D_DF, D_F, N_F, N_F, "-" )ANSWER( N_DF, D_F, D_DF, N_F, D_F, "+" )ANSWER( N_DF, D_DF, D_F, N_F, D_F, "+" )ANSWER( N_DF, D_F, D_DF, N_F, N_F, "+" )ANSWER( N_DF, D_DF, D_F, N_F, N_F, "+" )
Von der Kettenregel und der Produktregel wissen wir, dass \displaystyle \frac{\mathrm{d}}{\mathrm{d}x\strut}\frac{f(x)}{g(x)} = \frac{f'(x)g(x) - g'(x)f(x)}{g(x){}^2}.
In diesem Fall ist
\qquad f(x) = FUNCN.fText,
\qquad g(x) = FUNCD.fText.
Beide Funktionen ableiten:
\qquad f'(x) = FUNCN.ddxFText,
\qquad g'(x) = FUNCD.ddxFText.
Daher ist die Antwort
\qquad \dfrac{{(FUNCN.ddxFText)(FUNCD.fText) - (FUNCD.ddxFText)(FUNCN.fText)}}{(FUNCD.fText)^2}.