An Introduction to Finding the Derivative of cos(x) with the Chain Rule

d/dx(cosx)

To find the derivative of cos(x) with respect to x, we can use the derivative rules

To find the derivative of cos(x) with respect to x, we can use the derivative rules. The derivative of cos(x) can be found using the chain rule.

The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) is given by:

(d/dx) [f(g(x))] = f'(g(x)) * g'(x)

In our case, f(g(x)) is cos(x), and g(x) is x. Therefore, we need to find the derivatives of f(g(x)) and g(x) in order to apply the chain rule.

The derivative of g(x) = x is simply 1.

The derivative of f(g(x)) = cos(g(x)) involves the derivative of the cosine function. The derivative of cosine is given by the negative sine function. Therefore,

f'(g(x)) = -sin(g(x))

Now, we can apply the chain rule:

(d/dx) [cos(x)] = f'(g(x)) * g'(x)
= -sin(g(x)) * 1
= -sin(x)

Thus, the derivative of cos(x) with respect to x is -sin(x).

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