Question

Wyznaczanie z definicji pochodnej z wielomianu II stopnia, wyznaczanie z definicji pochodnej cząstkowej I rzędu - zad w załączniku

Answer 1

1.

[tex]\displaystyle f'(2)=\lim_{h\to 0}\dfrac{3(2+h)^2+5(2+h)+1-(3\cdot2^2+5\cdot2+1)}{h}\\\\ f'(2)=\lim_{h\to 0}\dfrac{3(4+4h+h^2)+10+5h+1-(12+10+1)}{h}\\\\ f'(2)=\lim_{h\to 0}\dfrac{12+12h+3h^2+5h+11-23}{h}\\\\ f'(2)=\lim_{h\to 0}\dfrac{3h^2+17h}{h}\\\\f'(2)=\lim_{h\to 0}(3h+17)\\\\f'(2)=3\cdot0+17=17[/tex]

2.

[tex]\displaystyle f_y'(2,1)=\lim_{h\to0}\dfrac{2^2+3\cdot2\cdot(1+h)^2+2\cdot(1+h)-(2^2+3\cdot2\cdot1^2+2\cdot1)}{h}\\\\f_y'(2,1)=\lim_{h\to0}\dfrac{4+6(1+2h+h^2)+2+2h-(4+6+2)}{h}\\\\f_y'(2,1)=\lim_{h\to0}\dfrac{4+6+12h+6h^2+2+2h-12}{h}\\\\f_y'(2,1)=\lim_{h\to0}\dfrac{6h^2+14h}{h}\\\\f_y'(2,1)=\lim_{h\to0}(6h+14)\\f_y'(2,1)=6\cdot0+14=14[/tex]