Understanding the Chain Rule | Derivative of sin(x) with respect to x = cos(x)

(d/dx) sin(x)

The notation (d/dx) represents differentiation with respect to the variable x

The notation (d/dx) represents differentiation with respect to the variable x. In this case, (d/dx) sin(x) denotes taking the derivative of the function sin(x) with respect to x.

To find the derivative of sin(x), we will use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then its derivative is given by the product of the derivative of the outer function (f'(g(x))) and the derivative of the inner function (g'(x)).

In this case, sin(x) is the outer function, and x is the inner function. The derivative of sin(x) with respect to x is found by following these steps:

1. Identify the outer function: sin(x).
2. Identify the inner function: x.
3. Apply the derivative of the outer function: The derivative of sin(x) is cos(x). This is a known result in calculus.
4. Apply the derivative of the inner function: The derivative of x with respect to x is 1.
5. Combine the derivatives: Multiply the derivative of the outer function (cos(x)) with the derivative of the inner function (1).
6. Simplify the derivative: The derivative of sin(x) with respect to x is cos(x).

So, (d/dx) sin(x) = cos(x).

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