How to Find the Derivative of the Sine Function | Explained with the Chain Rule and the Unit Circle

Derivative of sin(x)

The derivative of sin(x) with respect to x can be found using the chain rule and the derivative of the sine function

The derivative of sin(x) with respect to x can be found using the chain rule and the derivative of the sine function.

The derivative of the sine function, denoted as d/dx sin(x), is equal to the cosine function, or cos(x). So, d/dx sin(x) = cos(x).

To understand how this is derived, let’s consider the unit circle.

As a point P moves along the unit circle in the counterclockwise direction, the x-coordinate of P changes. Let’s call this x-coordinate as x, and let’s call the y-coordinate of P as y.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite to the angle to the hypotenuse. In the unit circle, the hypotenuse is of length 1, so the sine of an angle is equal to the y-coordinate of the corresponding point on the unit circle. Hence, sin(x) = y.

Now, let’s consider a small change in the angle. If we increase the angle by a small amount Δx, the new point on the unit circle can be denoted as P’. The x-coordinate of P’ is (x + Δx), and the y-coordinate is y’.

The change in y-coordinate, Δy, can be written as y’ – y. We can find an approximation for Δy by considering the triangle formed by the points P and P’. The change in x-coordinate, Δx, is equal to the length of the adjacent side of the triangle.

Now, by applying the approximation for small angles, we have sin(Δx) ≈ Δy, and cos(Δx) ≈ Δx. Dividing both sides by Δx, we get sin(Δx)/Δx ≈ Δy/Δx. As Δx approaches 0, this approximation becomes more accurate.

Taking the limit as Δx approaches 0, we obtain the derivative of sin(x) as d/dx sin(x) = cos(x).

In conclusion, the derivative of the sine function, sin(x), with respect to x is equal to the cosine function, cos(x).

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