# Find the derivative of -\sin(x)

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

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

### Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Juan Spiller
$$\displaystyle{f}'{\left({x}\right)}=\lim_{{{h}\rightarrow{0}}}{\frac{{{\sin{{\left({x}+{h}\right)}}}-{\sin{{\left({x}\right)}}}}}{{{h}}}}$$
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 $$\displaystyle{f}'{\left({x}\right)}={\cos{{\left({x}\right)}}}$$
###### Not exactly what youâ€™re looking for?
Foreckije
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{\sin{{\left({x}\right)}}}\right)}$$
Take the constant out
$$\displaystyle-{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}$$
Apply the common derivative:
$$\displaystyle=-{\cos{{\left({x}\right)}}}$$