The Derivative of Sin x | Understanding the Relationship Between Sine and Cosine

The derivative of sin x is…

The derivative of sin x is cos x

The derivative of sin x is cos x.

To understand why, let’s start with the definition of the derivative. The derivative of a function represents the rate of change of that function at any given point. In other words, it shows how the function is changing as we move along the x-axis.

In the case of sin x, we can start by approximating the derivative using the concept of limits. We consider a small change in x, denoted as Δx, and examine the ratio of the change in the function value (Δy) to the change in x:

Δy/Δx = (sin(x + Δx) – sin(x))/Δx

Now, if we want to find the derivative, we need to take the limit as Δx approaches 0:

lim(Δx→0) (sin(x + Δx) – sin(x))/Δx

Using the identity for the difference of two sines:

sin(A) – sin(B) = 2cos((A+B)/2)sin((A-B)/2)

We can rewrite our expression as:

lim(Δx→0) [2cos((x + Δx + x)/2)sin((x + Δx – x)/2)]/Δx

Simplifying further:

lim(Δx→0) [2cos((2x + Δx)/2)sin(Δx/2)]/Δx

Now, we can simplify the expression in the limit:

lim(Δx→0) [2cos(x + Δx/2)sin(Δx/2)]/Δx

As Δx approaches 0, sin(Δx/2) becomes approximately equal to Δx/2, and cos(x + Δx/2) approaches cos(x). Thus, we can rewrite the expression as:

lim(Δx→0) [2cos(x) * (Δx/2)]/Δx

Canceling Δx:

lim(Δx→0) cos(x)

Finally, as Δx approaches 0, the limit becomes:

cos(x)

Therefore, the derivative of sin x is cos x.

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