Exploring the Chain Rule: The Derivative of Cosine of x

derivative of cosx

To find the derivative of the function cosine of x (cos x), we can use the basic rules of differentiation

To find the derivative of the function cosine of x (cos x), we can use the basic rules of differentiation.

The derivative of cos x with respect to x can be found using the chain rule. The chain rule states that if we have a composite function, f(g(x)), the derivative can be obtained by multiplying the derivative of the outer function by the derivative of the inner function.

In this case, our function f(x) is cos x. The derivative of cos x, denoted as d/dx (cos x), can be obtained as follows:

First, we need to recall the derivative of the sine function (sin x), as cosine is related to sine:

d/dx (sin x) = cos x

Now let’s use the chain rule. We can write cos x as f(g(x)), where f(u) = cos u and g(x) = x.

Using the chain rule, we have:

d/dx (cos x) = d/du (cos u) * d/dx (x)

Since u = x, we can simplify this to:

d/dx (cos x) = d/du (cos u) * d/dx (u)

Next, we need to find the derivative of the outer function, f(u) = cos u, with respect to its variable, u.

d/du (cos u) = -sin u

Now we substitute this value back into our chain rule equation:

d/dx (cos x) = -sin u * d/dx (u)

Since u = x, we can rewrite this as:

d/dx (cos x) = -sin x * d/dx (x)

Finally, the derivative of x with respect to x is simply 1. Therefore, the derivative of the function cos x with respect to x is:

d/dx (cos x) = -sin x

So, the derivative of cos x is minus sine of x, or -sin x.

More Answers: