Understanding the Derivative of Cos(x) | A Step-by-Step Explanation

derivative of cosx

The derivative of cos(x) is given by the following formula:

d/dx[cos(x)] = -sin(x)

To understand how this derivative is derived, we can use the definition of the derivative

The derivative of cos(x) is given by the following formula:

d/dx[cos(x)] = -sin(x)

To understand how this derivative is derived, we can use the definition of the derivative. The derivative of a function f(x) is the rate at which the function is changing at a particular point.

In this case, we consider the cosine function, which gives us the ratio of the adjacent side of a right triangle to its hypotenuse for a given angle. The derivative of the cosine function gives us the rate at which this ratio is changing as the angle changes.

Using trigonometric identities, we can express the cosine function as follows:

cos(x) = (e^(ix) + e^(-ix)) / 2

Now, let’s find the derivative of cos(x) using the definition of the derivative:

d/dx[cos(x)] = lim(h->0) [cos(x + h) – cos(x)] / h

Expanding the terms, we have:

= lim(h->0) [(e^(i(x + h)) + e^(-i(x + h))) / 2 – (e^(ix) + e^(-ix)) / 2] / h

Simplifying further, we get:

= lim(h->0) [e^ix * e^ih + e^(-ix) * e^(-ih) – e^ix – e^(-ix)] / (2h)

Using Euler’s formula, which states e^(ix) = cos(x) + i*sin(x), we can rewrite the expression as:

= lim(h->0) [(cos(x) + i*sin(x)) * (cos(h) + i*sin(h)) + (cos(x) – i*sin(x)) * (cos(-h) + i*sin(-h)) – cos(x) – cos(-x)] / (2h)

Simplifying further and canceling out the terms, we obtain:

= lim(h->0) [(cos(x)*cos(h) – sin(x)*sin(h)) + i*(sin(x)*cos(h) + cos(x)*sin(h)) – cos(x) – cos(-x)] / (2h)

Now, focus on the real part of the expression:

= lim(h->0) [(cos(x)*cos(h) – sin(x)*sin(h)) – cos(x)] / (2h)

Using the trigonometric identity cos(x + y) = cos(x)*cos(y) – sin(x)*sin(y), we have:

= lim(h->0) [cos(x + (-h)) – cos(x)] / (2h)

Taking the limit as h approaches 0, the expression becomes:

= -sin(x)

Therefore, the derivative of cos(x) is -sin(x).

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