Mathematrix: Aufgabenbeispiele/ Ableitung und Grenzwerten

01

01}}

$\frac{d(y)}{dx}=\frac{d(a{x}^4)}{dx}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }(a{x}^4)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{a(x+\mathrm{\Delta }x{)}^4-a\mathrm{\Delta }{x}^4}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{a(x^4+4x^3\mathrm{\Delta }x+6x^2\mathrm{\Delta }{x}^2+4x\mathrm{\Delta }{x}^3+\mathrm{\Delta }{x}^4)-a{x}^4}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\cancel{a{x}^4}+4a{x}^3\mathrm{\Delta }x+6a{x}^2\mathrm{\Delta }{x}^2+4ax\mathrm{\Delta }{x}^3+a\mathrm{\Delta }{x}^4-\cancel{a{x}^4}}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\cancel{\mathrm{\Delta }x}(4a{x}^3+6a{x}^2\mathrm{\Delta }x+4ax\mathrm{\Delta }{x}^2+a\mathrm{\Delta }{x}^3)}{\cancel{\mathrm{\Delta }x}}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }(4a{x}^3+6a{x}^2{\xcancel{\mathrm{\Delta }x}}^0+4ax{\xcancel{\mathrm{\Delta }{x}^2}}^0+a{\xcancel{\mathrm{\Delta }{x}^3}}^0)=4a{x}^3$

Also: $\frac{d(a{x}^4)}{dx}=4a{x}^3$

02

02}}

$\frac{d(y)}{dx}=\frac{d(\frac{2}{5}{x}^5)}{dx}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }(\frac{2}{5}{x}^5)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{2}{5}(x+\mathrm{\Delta }x{)}^5-\frac{2}{5}\mathrm{\Delta }{x}^5}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{2}{5}(x^5+5x^4\mathrm{\Delta }x+10x^3\mathrm{\Delta }{x}^2+10x^2\mathrm{\Delta }{x}^3+5x\mathrm{\Delta }{x}^4+\mathrm{\Delta }{x}^5)-\frac{2}{5}{x}^5}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\cancel{\frac{2}{5}{x}^5}+\frac{2}{5}5x^4\mathrm{\Delta }x+\frac{2}{5}10x^3\mathrm{\Delta }{x}^2+\frac{2}{5}10x^2\mathrm{\Delta }{x}^3+\frac{2}{5}5x\mathrm{\Delta }{x}^4+\frac{2}{5}\mathrm{\Delta }{x}^5-\cancel{\frac{2}{5}{x}^5}}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{2}{5}(5x^4\mathrm{\Delta }x+10x^3\mathrm{\Delta }{x}^2+10x^2\mathrm{\Delta }{x}^3+5x\mathrm{\Delta }{x}^4+\mathrm{\Delta }{x}^5)}{\mathrm{\Delta }x}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\frac{2}{5}\cancel{\mathrm{\Delta }x}(5x^4+10x^3\mathrm{\Delta }x+10x^2\mathrm{\Delta }{x}^2+5x\mathrm{\Delta }{x}^3+\mathrm{\Delta }{x}^4)}{\cancel{\mathrm{\Delta }x}}=$

$\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{2}{5}(5x^4+10x^3{\xcancel{\mathrm{\Delta }x}}^0+10x^2{\xcancel{\mathrm{\Delta }{x}^2}}^0+5x{\xcancel{\mathrm{\Delta }{x}^3}}^0+{\xcancel{\mathrm{\Delta }{x}^4}}^0)=\frac{2}{5}5x^4=2x^4$

Also: $\frac{d(\frac{2}{5}{x}^5)}{dx}=\frac{2}{5}\cdot 5x^4=2x^4$

03

03}}

$\frac{d(y)}{dx}=\frac{d(ax)}{dx}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{\mathrm{\Delta }(ax)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{a(x+\mathrm{\Delta }x)-ax}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{a\cancel{\mathrm{\Delta }x}}{\cancel{\mathrm{\Delta }x}}=a$

Also: $\frac{d(ax)}{dx}=a$