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

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

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

The chain rule states that if we have a composition of functions, g(f(x)), the derivative of the composition can be found by multiplying the derivative of the outer function g'(f(x)) with the derivative of the inner function f'(x).

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

The derivative of the outer function g(x) = sin(x) can be found using the derivative of the sine function, which is cos(x). So, g'(x) = cos(x).

The derivative of the inner function f(x) = x is simply 1, as the derivative of x with respect to x is 1.

Now, we can apply the chain rule to find the derivative of f(x) = g(f(x)):
f'(x) = g'(f(x)) * f'(x)
= cos(f(x)) * 1
= cos(x)

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

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