Question Number 177570 by mathlove last updated on 07/Oct/22
$$\underset{\Delta{x}\rightarrow{cos}\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{{sin}^{\mathrm{3}} \left(\Delta{x}+{x}\right)−{sin}^{\mathrm{3}} {x}}{\mathrm{2}^{−\mathrm{1}} \Delta{x}}=? \\ $$
Commented by blackmamba last updated on 07/Oct/22
$$ \\ $$$$\mathrm{cos}\:\frac{\pi}{\mathrm{2}}=\mathrm{0}? \\ $$
Commented by blackmamba last updated on 07/Oct/22
$$\:\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{3}} \left({x}+\Delta{x}\right)−\mathrm{sin}\:^{\mathrm{3}} {x}}{\frac{\mathrm{1}}{\mathrm{2}}\Delta{x}}\:? \\ $$
Commented by mathlove last updated on 07/Oct/22
$$\Delta{x}\rightarrow{cos}\frac{\pi}{\mathrm{2}} \\ $$
Commented by mathlove last updated on 07/Oct/22
$${yes} \\ $$
Commented by mathlove last updated on 07/Oct/22
$${any}\:{one}\:{solve}\:{this}\:{qustion} \\ $$
Answered by mr W last updated on 07/Oct/22
$$\underset{\Delta{x}\rightarrow{cos}\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{{sin}^{\mathrm{3}} \left(\Delta{x}+{x}\right)−{sin}^{\mathrm{3}} {x}}{\mathrm{2}^{−\mathrm{1}} \Delta{x}} \\ $$$$=\mathrm{2}\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{sin}^{\mathrm{3}} \left(\Delta{x}+{x}\right)−{sin}^{\mathrm{3}} {x}}{\Delta{x}} \\ $$$$=\mathrm{2}×\left(\mathrm{sin}^{\mathrm{3}} \:{x}\right)' \\ $$$$=\mathrm{6}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:\mathrm{cos}\:{x} \\ $$
Commented by mathlove last updated on 08/Oct/22
$${thanks}\:{a}\:{lot}\:{sir} \\ $$