The Chain Rule: Finding the Derivative of sin(f(x)) with Respect to x

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

To find the derivative of sin(f(x)) with respect to x, we can use the chain rule of differentiation

To find the derivative of sin(f(x)) with respect to x, we can use the chain rule of differentiation.

The chain rule states that if we have a composition of functions, such as sin(f(x)), where f(x) is the inner function, and g(x) is the outer function, then the derivative of the composition is given by:

(d/dx)[g(f(x))] = g'(f(x)) * f'(x)

In this case, the outer function is sin(x) and the inner function is f(x).

First, let’s find the derivative of the outer function, g'(x) = d/dx[sin(x)]:

Using the derivative of sin(x), we know that d/dx[sin(x)] = cos(x).

Now, let’s find the derivative of the inner function, f'(x):

This part is not specified in the given problem, so we cannot determine the exact derivative of f(x) without more information. If you have more information about the specific function f(x), please provide it, and we can continue solving the problem.

However, assuming that f(x) is a differentiable function, we can represent its derivative as f'(x), and the overall derivative can be written as:

(d/dx)[sin(f(x))] = cos(f(x)) * f'(x)

So, the derivative of sin(f(x)), with respect to x, is equal to cos(f(x)) times the derivative of the inner function f(x), given by f'(x).

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