Question

$\huge{\mathcal{\underline{\pink{Hey!\:Dudes}}}}$
❥Find the derivative of sinx. ​

Answer 1

Answer:

Cos x

Step-by-step explanation:

The derivative of sin x

d/dx sin x = cos x

To prove that, we will apply the definition of the derivative. First, we will calculate the difference quotient.

Derivative of sin x = Derivative of sin x , Problem 1,

= Derivative of sin x , on dividing numerator

and denominator by 2,

= Derivative of sin x

We will now take the limit as h Right arrow0. But the limit of a product is equal to the product of the limits. (Lesson 2.) And the factor on the right has the form sin θ/θ. Therefore, according to the Lemma, as h Right arrow0 its limit is 1. Therefore,

d

dx sin x = cos x.

We have established the formula.

Answer 2

$\Large\underline\mathfrak{Question}$

• Find the Derivative of sinx . ?

$\underline {\underline{\LARGE{{\bf{\green{S}}}{\mathfrak{o}}{\mathfrak{\orange{l}}}{\mathfrak{\red{u}}}{\mathfrak{\pink{t}}}{\mathfrak{\purple{i}}}{\mathfrak{\blue{o}}}{\mathfrak{\red{n}}}}}} : \:$

$\textsf{To Prove this , we use the limit Definition of Derivative .} \\ </p><p> \textbf{Limit Definition for sin:-}$

$\red{ \boxed{ \bf \frac{d}{dx} \sin(x) = \displaystyle \lim_{h \to 0} \frac{sin(x + h) - sin(x)}{h}}}$

$\sf \: now \: using \: trignometry \: angle \: sum \: formula : - \\ \\ \boxed{ \bf \: sinx - siny = 2sin (\frac{x - y}{2} )cos( \frac{x + y}{2} )}$

$\red\longrightarrow \bf \frac{d}{dx} \sin(x) = \displaystyle \lim_{h \to 0} \: \frac{2sin \frac{h}{2}cos(x + \frac{h}{2} }{h} \\ \\ \red\longrightarrow \bf \frac{d}{dx} \sin(x) = \displaystyle \lim_{h \to 0} \: \frac{2sin \frac{h}{2} }{h} \times \displaystyle \lim_{h \to 0} \: \frac{cos(x + \frac{h}{2}) }{h} \\ \\ \sf \: now \: we \: know \: that \: \\ \green{\boxed{\displaystyle \lim_{h \to 0} \: \dfrac{sin(h)}{h} = 1}} \\ \\ \red\longrightarrow \bf \frac{d}{dx} \sin(x) \: = 1 \times cos(x) \\ \\ \red\longrightarrow \bf \pink{\large\boxed{\boxed{\bold{ \frac{d}{dx} \sin(x) \: = cos(x)}}}}$

_________________________________

Hence, the Derivate of sin(x) is cos(x).

$\large\underline\textbf{Hope it Helps You.}$

#BAL