Differentiation of sinx

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

Let’s begin –

Differentiation of sinx

The differentiation of sinx with respect to x is cosx.

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

Proof Using First Principle :

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

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

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

By using trigonometry formula,

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

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

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

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

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

\(\implies\) \(d\over dx\)(f(x)) = (cos x) \(\times\) 1 = cos x

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

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

Solution : Let y = sin 2x – 2 sin x 

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

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

By using chain rule and sinx differentiation we get,

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

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

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

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

Solution : Let y = \(x^2\) +  sin x

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

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

By using differentiation formulas we get,

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

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


Related Questions

What is the Differentiation of sin inverse x ?

What is the differentiation of \(sin x^2\) ?

What is the Integration of sin x ?

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