How to Find the Derivative of the Cosine Function | Understanding and Using the Chain Rule.

d/dx cos x

To find the derivative of the function f(x) = cos(x), we can use the chain rule and the derivative of the cosine function

To find the derivative of the function f(x) = cos(x), we can use the chain rule and the derivative of the cosine function.

The chain rule states that if we have a composite function g(f(x)), then its derivative is given by the product of the derivative of the outer function g'(f(x)) and the derivative of the inner function f'(x).

In this case, our function f(x) = cos(x) can be thought of as the composite function g(u) = cos(u), where u = x. So we need to find the derivative of g(u) with respect to u, and then multiply it by the derivative of u with respect to x.

The derivative of g(u) = cos(u) with respect to u is given by g'(u) = -sin(u). This is a well-known derivative of the cosine function.

The derivative of u = x with respect to x is simply 1.

Now, using the chain rule, we can find the derivative of f(x) = cos(x) as follows:

f'(x) = g'(u) * 1 (chain rule)
= -sin(u) * 1 (substituting the values of g'(u) and 1)
= -sin(x) (substituting the value of u = x)

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

In summary, d/dx(cos(x)) = -sin(x).

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