Understanding the Chain Rule: Finding the Derivative of the Cosine Function

Derivative of cos(x)

The derivative of the cosine function, written as d(cos(x))/dx or (d/dx) cos(x), represents the rate of change of the cosine function with respect to the variable x

The derivative of the cosine function, written as d(cos(x))/dx or (d/dx) cos(x), represents the rate of change of the cosine function with respect to the variable x.

To find the derivative of cos(x), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative of this function can be found by taking the derivative of the outer function with respect to the inner function multiplied by the derivative of the inner function with respect to x.

In the case of the cosine function, we can think of it as a composition of two functions: f(u) = cos(u) and g(x) = x. So, f(g(x)) = cos(x).

Using the chain rule, we need to find the derivatives of both the outer and inner functions. The derivative of the outer function, d(cos(u))/du, is simply given by -sin(u). And the derivative of the inner function, d(x)/dx, is equal to 1.

Now, we can apply the chain rule:

(d/dx) cos(x) = (d/dx) f(g(x)) = (d/dx) cos(u) * (du/dx) = -sin(u) * 1 = -sin(x).

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

In summary,

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

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