Stacie Worsley

2021-12-14

Find the derivative of $-\mathrm{sin}\left(x\right)$

Juan Spiller

${f}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{\mathrm{sin}\left(x+h\right)-\mathrm{sin}\left(x\right)}{h}$
Let's use representation of a difference of sin functions as a product of sin and cos, and we get:
${f}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{2×\mathrm{sin}\left(\frac{h}{2}\right)\mathrm{cos}\left(x+\frac{h}{2}\right)}{h}$
${f}^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{\mathrm{sin}\left(\frac{h}{2}\right)}{\frac{h}{2}}×\underset{h\to 0}{lim}\mathrm{cos}\left(x+\frac{h}{2}\right)$
${f}^{\prime }\left(x\right)=1×\mathrm{cos}\left(x\right)=\mathrm{cos}\left(x\right)$
Thus, the derivative is ${f}^{\prime }\left(x\right)=\mathrm{cos}\left(x\right)$

Foreckije

$\frac{d}{dx}\left(-\mathrm{sin}\left(x\right)\right)$
Take the constant out
$-\frac{d}{dx}\left(\mathrm{sin}\left(x\right)\right)$
Apply the common derivative:
$=-\mathrm{cos}\left(x\right)$

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