Differentiation of cosx

Here you will learn what is the differentiation of cosx and its proof by using first principle.

Let’s begin –

Differentiation of cosx

The differentiation of cosx with respect to x is -sinx.

i.e. \(d\over dx\) (cosx) = -sinx

Proof Using First Principle :

Let f(x) = cos x. Then, f(x + h) = cos(x + h)

\(\therefore\)   \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(f(x + h) – f(x)\over h\)

\(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(cos(x + h) – cos x\over h\)

By using trigonometry formula,

[cos C – cos D = \(-2sin{C + D\over 2}sin{C – D\over 2}\)]

\(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(-2sin({h\over 2})sin({{2x + h}\over 2})\over h\)

\(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(-2sin({h/2})sin({{x + h/2}\over 2})\over 2(h/2)\)

\(\implies\) \(d\over dx\)(f(x)) = -\(lim_{h\to 0}\) \(sin({{x + h/2}\over 2})\) \(lim_{h\to 0}\)\(sin(h/2)\over (h/2)\)

because, [\(lim_{h\to 0}\)\(sin(h/2)\over (h/2)\) = 1]

\(\implies\) \(d\over dx\)(f(x)) = -(sin x) \(\times\) 1 = – sin x

Hence, \(d\over dx\) (cos x) = -sin x

Example : What is the differentiation of cos x – 2 sin x with respect to x?

Solution : Let y = cos x – 2 sin x 

\(d\over dx\)(y) = \(d\over dx\)(cos x – 2 sin x)

\(\implies\) \(d\over dx\)(y) = \(d\over dx\)(cos x) – \(d\over dx\)(2 sinx)

By using cosx and sinx differentiation we get,

\(\implies\) \(d\over dx\)(y) = -sin x – 2 \(d\over dx\)(sinx)

\(\implies\) \(d\over dx\)(y) = -sin x – 2 cos x

Hence, \(d\over dx\)(cos x – 2 sin x) = -sin x – 2 cos x

Example : What is the differentiation of \(x^2\) +  cos 2x with respect to x?

Solution : Let y = \(x^2\) + cos 2x

\(d\over dx\)(y) = \(d\over dx\)(\(x^2\) +  cos 2x)

\(\implies\) \(d\over dx\)(y) = \(d\over dx\)\(x^2\) – \(d\over dx\)(cos 2x)

By using chain rule and differentiation formulas we get,

\(\implies\) \(d\over dx\)(y) = 2x + (2)(-sin 2x)

Hence, \(d\over dx\)(\(x^2\) +  cos 2x) = 2x – 2sin 2x


Related Questions

What is the Differentiation of cos inverse x ?

What is the differentiation of cosx sinx ?

What is the Integration of cos x ?

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