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Question Number 216014 by mathlove last updated on 25/Jan/25

$$\underset{\mathrm{\Delta }x\to cos\frac{\pi }{2}}{\lim }\frac{{\sin }^3(\mathrm{\Delta }x+x)-{sin}^{3}x}{2^{-1}\cdot \mathrm{\Delta }x}=?$$

Answered by MrGaster last updated on 26/Jan/25

$$=2\cdot \underset{\mathrm{\Delta }x\to 0}{\lim }\frac{{\sin }^3(\mathrm{\Delta }x-x)-{\sin }^{3}x}{\mathrm{\Delta }x}$$$$=2\cdot \frac{d}{dx}{\sin }^{3}x$$$$=2\cdot {3\sin }^{2}x\cdot \cos x$$$$={6\sin }^{2}x\cos x$$

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