How to Find the Derivative of cos(x) Using the Chain Rule in Calculus

(d/dx) cos(x)

The expression (d/dx) cos(x) represents the derivative of the function cos(x) with respect to x

The expression (d/dx) cos(x) represents the derivative of the function cos(x) with respect to x.

To calculate this derivative, we can use the chain rule. The chain rule states that if we have a composition of two functions, u(v(x)), the derivative of this composition is given by (du/dv) * (dv/dx).

In our case, the function u(x) = cos(x) and v(x) = x. Therefore, the derivative of cos(x) with respect to x is the derivative of u(v(x)) with respect to x, which is (du/dv) * (dv/dx).

First, let’s find the derivatives of u(x) and v(x):
– The derivative of u(x) = cos(x) with respect to v is du/dv = -sin(v).
– The derivative of v(x) = x with respect to x is dv/dx = 1.

Now, we can substitute these derivatives into the chain rule:
(d/dx) cos(x) = (du/dv) * (dv/dx) = -sin(v) * 1

Since v = x, we have:
(d/dx) cos(x) = -sin(x).

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

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