The Chain Rule Explained: Derivative of cos(x) using d/dx

d/dx cos(x)

To find the derivative of the function f(x) = cos(x) using the derivative operator d/dx, we can apply the chain rule

To find the derivative of the function f(x) = cos(x) using the derivative operator d/dx, we can apply the chain rule. The chain rule tells us that if we have a composite function, f(g(x)), then its derivative with respect to x is given by f'(g(x)) * g'(x).

In this case, our composite function is f(g(x)) = cos(x), where g(x) = x. So, we can rewrite f(g(x)) as cos(g(x)).

Now, let’s apply the chain rule. The derivative of f(g(x)) with respect to x is given by:

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

In this case, g(x) = x, so d/dx [g(x)] = d/dx [x] = 1.

Substituting this back into our derivative expression, we get:

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

Therefore, the derivative of f(x) = cos(x) with respect to x is -sin(x).

So, if you take the derivative of cos(x) with respect to x (d/dx [cos(x)]), you get -sin(x).

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