Differentiation of sinx

The differentiation of (\sin(x)) with respect to (x) is a fundamental concept in calculus. The derivative of (\sin(x)) is given by:

\[
\frac{d}{dx}(\sin(x)) = \cos(x).
\]

This result is derived from the limit definition of the derivative and is a key part of the differentiation rules for trigonometric functions. The cosine function, $\cos(x)$, represents the rate of change of $\sin(x)$ with respect to (x).

Proof of Formula

We can proof using the below two formulas

$\frac {d}{dx} f(x) =\displaystyle \lim_{h \to 0} \frac {f (x+h) – f(x)}{h}$

$\displaystyle \lim_{x \to 0} \frac {sinx}{x}=1$

Now

$\frac {d}{dx} (sin x) = \displaystyle \lim_{h \to 0} \frac {sin (x+h) – sin x}{h}= \displaystyle \lim_{h \to 0} \frac {2cos (\frac {2x+h}{2}) sin \frac {h}{2}}{h}$

$=\displaystyle \lim_{h \to 0} cos (x + \frac {h}{2}) \displaystyle \lim_{h \to 0} \frac {sin \frac {h}{2}}{\frac {h}{2}} = cos x$

as $\displaystyle \lim_{x \to 0} \frac {sinx}{x}=1$

Domain of the Derivative of sinx

\[
\frac{d}{dx}(\sin(x)) = \cos(x).
\]

We can clearly see that Domain of the Derivative of sin x is R

Geometrical Meaning of the Derivative

The geometric meaning of the derivative of $\sin(x)$, which is $\cos(x)$, can be understood by examining the graph of $\sin(x)$ and considering what the derivative represents.

1. Slope of the Tangent Line: The derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point. For sin(x), the derivative at any point (x) is \cos(x). This means that the slope of the tangent line to the curve y = sin(x) at any point (x) is equal to cos(x).
2. Rate of Change: Geometrically, cos(x) represents the rate of change of $\sin(x)$ with respect to (x). When $\sin(x)$ is increasing (as in the first quadrant of the unit circle, where $0 < x < \pi/2$, its derivative, $\cos(x)$, is positive. When $\sin(x)$ is decreasing (as in the second quadrant, where $\pi/2 < x < \pi$, $\cos(x)$ is negative.
3. Maximum and Minimum Points: At the points where (\sin(x)) reaches its maximum $x = \pi/2 + 2k\pi$ or minimum $x = 3\pi/2 + 2k\pi$, the derivative $\cos(x)$ is zero. This corresponds to the peaks and troughs of the sine wave, where the tangent line is horizontal.
4. Harmonic Motion: In physics, $\sin(x)$ often represents simple harmonic motion, such as the displacement of a pendulum or a mass on a spring. The derivative, $\cos(x)$, then represents the velocity of the motion, showing how quickly and in what direction the displacement is changing at any point in time.

Solved Questions

Question 1:
Differentiate $y = \sin(3x)$.
Solution:
To differentiate $y = \sin(3x)$, we use the chain rule. Let (u = 3x), then $y = \sin(u)$. Differentiating (y) with respect to (x):
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot 3 = 3\cos(3x).
\]

Question 2:
Find the derivative of $y = x \sin(x)$.
Solution
Here, we use the product rule. If (y = uv) where (u = x) and $v = \sin(x)$, then (y’ = u’v + uv’). Therefore:

\[
\frac{dy}{dx} = 1 \cdot \sin(x) + x \cdot \cos(x) = \sin(x) + x\cos(x).
\]

Question 3:
What is the second derivative of $y = \sin(x)$?
Solution:
First, we find the first derivative, which is $y’ = \cos(x)$. Then, differentiating again:

\[
y” = \frac{d}{dx}(\cos(x)) = -\sin(x).
\]