How to Find the Derivative of cos(f(x)) Using the Chain Rule: Step-by-Step Guide

d/dx [ cos(f(x)) ]

To find the derivative of cos(f(x)), where f(x) is a function, we can use the chain rule

To find the derivative of cos(f(x)), where f(x) is a function, we can use the chain rule.

The chain rule states that if we have a composite function y = f(g(x)), then the derivative of y with respect to x (dy/dx) can be found by taking the derivative of the outer function f'(g(x)) and multiplying it by the derivative of the inner function g'(x).

In this case, our composite function is y = cos(f(x)), where the outer function is cos(u) and the inner function is f(x).

Step 1: Find the derivative of the outer function cos(u).
The derivative of cos(u) is -sin(u). Note that we treat u as a variable here, as it represents the input to cos(u).

Step 2: Find the derivative of the inner function f'(x).
This depends on the specific form of f(x). If you provide the specific expression for f(x), I can guide you in finding its derivative.

Once we have the derivative of the inner function, we can apply the chain rule.

Step 3: Apply the chain rule.
Multiplying the derivative of the outer function (-sin(u)) by the derivative of the inner function (f'(x)), we get:
dy/dx = (-sin(f(x))) * f'(x)

Therefore, the derivative of cos(f(x)) is (-sin(f(x))) * f'(x), where f'(x) represents the derivative of the inner function f(x).

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