Understanding the Chain Rule: Finding the Derivative of cos(x) with Respect to x

d/dx(cosx)

To find the derivative of cos(x) with respect to x (d/dx (cos(x))), we can use the chain rule

To find the derivative of cos(x) with respect to x (d/dx (cos(x))), we can use the chain rule.

The chain rule states that for a composition of functions, if y = f(g(x)), then the derivative dy/dx can be found by taking the derivative of the outer function f’ with respect to its inner function multiplied by the derivative of the inner function g’ with respect to x.

In this case, we have y = cos(x), where f(u) = cos(u) and g(x) = x. So, applying the chain rule:

dy/dx = f'(g(x)) * g'(x)

We find the derivative of f(u) = cos(u) with respect to u, which is obtained by differentiating cosine:

f'(u) = -sin(u)

Now, we substitute u with g(x):

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

Finally, we find the derivative of g(x) = x, which is simply 1:

g'(x) = 1

So, putting it all together:

dy/dx = f'(g(x)) * g'(x) = -sin(g(x)) * 1

Since g(x) = x:

dy/dx = -sin(x)

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

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