Derivative of Cos(x) with Respect to x: The Chain Rule Simplified

(d/dx) cos(x)

To find the derivative of the function f(x) = cos(x), with respect to x, we can use the chain rule

To find the derivative of the function f(x) = cos(x), with respect to x, we can use the chain rule. The chain rule states that if we have a function g(u) and another function f(x), where x = g(u), then the derivative of f(x) with respect to x is given by:

(d/dx) f(x) = (df/du) * (du/dx)

Applying the chain rule to our function f(x) = cos(x), we can rewrite it as f(u) = cos(u), where u = x. So, we have:

f(u) = cos(u)

Now, we need to find the derivative of f(u) with respect to u, or in other words, we need to find (df/du).

(d/dx) f(x) = (df/du) * (du/dx)

If f(u) = cos(u), then (df/du) is the derivative of cos(u) with respect to u, which is given by:

(df/du) = -sin(u)

Next, we need to find (du/dx). Since u = x in our case, (du/dx) is simply 1.

So, we can rewrite the derivative of f(x) as:

(d/dx) f(x) = (df/du) * (du/dx) = -sin(u) * 1

Substituting back u = x, we get:

(d/dx) f(x) = -sin(x)

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

More Answers: