Find the derivative of -\sin(x)

Stacie Worsley 2021-12-14 Answered
Find the derivative of \(\displaystyle-{\sin{{\left({x}\right)}}}\)

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Expert Answer

Juan Spiller
Answered 2021-12-15 Author has 6262 answers
\(\displaystyle{f}'{\left({x}\right)}=\lim_{{{h}\rightarrow{0}}}{\frac{{{\sin{{\left({x}+{h}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{h}}}}\)
Let's use representation of a difference of sin functions as a product of sin and cos, and we get:
\(\displaystyle{f}'{\left({x}\right)}=\lim_{{{h}\rightarrow{0}}}{\frac{{{2}\times{\sin{{\left({\frac{{{h}}}{{{2}}}}\right)}}}{\cos{{\left({x}+{\frac{{{h}}}{{{2}}}}\right)}}}}}{{{h}}}}\)
\(\displaystyle{f}'{\left({x}\right)}=\lim_{{{h}\rightarrow{0}}}{\frac{{{\sin{{\left({\frac{{{h}}}{{{2}}}}\right)}}}}}{{{\frac{{{h}}}{{{2}}}}}}}\times\lim_{{{h}\rightarrow{0}}}{\cos{{\left({x}+{\frac{{{h}}}{{{2}}}}\right)}}}\)
\(\displaystyle{f}'{\left({x}\right)}={1}\times{\cos{{\left({x}\right)}}}={\cos{{\left({x}\right)}}}\)
Thus, the derivative is \(\displaystyle{f}'{\left({x}\right)}={\cos{{\left({x}\right)}}}\)
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Foreckije
Answered 2021-12-16 Author has 1034 answers
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{\sin{{\left({x}\right)}}}\right)}\)
Take the constant out
\(\displaystyle-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}\)
Apply the common derivative:
\(\displaystyle=-{\cos{{\left({x}\right)}}}\)
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