How to Find the Derivative of Cos(x) Using Chain Rule of Differentiation

d/dx(cosx)

sinx

To find the derivative of cos(x) with respect to x, we need to use the chain rule of differentiation.

The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

In this case, we can let f(u) = cos(u) and g(x) = x. Therefore, cos(x) = f(g(x)) = f(x).

To find the derivative of cos(x), we need to find the derivative of f(x) with respect to g(x) and then multiply it with the derivative of g(x) with respect to x.

So,

f(u) = cos(u)

f'(u) = -sin(u) (using derivative of cos(u) = -sin(u))

g(x) = x

g'(x) = 1

Now, applying the chain rule:

(d/dx) cos(x) = f'(g(x)) * g'(x)

= f'(x) * 1

= -sin(x)

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

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