Finding the Derivative of Cos x Using the Chain Rule | Simplifying d/dx Cos x

d/dx cos x

The expression “d/dx” represents the derivative with respect to x

The expression “d/dx” represents the derivative with respect to x. So, when we say “d/dx cos x,” we are asking to find the derivative of the cosine function with respect to 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, g(f(x)), then the derivative of that composite function with respect to x is given by the product of the derivative of the outer function g'(f(x)) with the derivative of the inner function f'(x).

In this case, the outer function is just the cosine function, and the inner function is x itself. The derivative of the inner function, x, with respect to x is simply 1.

Now, let’s find the derivative of the outer function, cos(x). The derivative of cos(x) is given by -sin(x). Therefore, g'(f(x)) = -sin(x).

Since f(x) = x and its derivative, f'(x) = 1, the chain rule tells us that the derivative of cos(x) with respect to x, which is d/dx cos(x), is:

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

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

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