How to Calculate the Derivative of sin(x) Using the Chain Rule

d/dx sinx

To find the derivative of the function f(x) = sin(x) with respect to x, denoted as d/dx sin(x), we will use the chain rule

To find the derivative of the function f(x) = sin(x) with respect to x, denoted as d/dx sin(x), we will use the chain rule.

The chain rule states that if we have a composite function g(f(x)), where g(x) and f(x) are differentiable, then the derivative of g(f(x)) with respect to x is given by g'(f(x)) * f'(x).

In this case, the function sin(x) can be viewed as the composition of two functions: f(x) = sin(x) and g(x) = x. Applying the chain rule, we have:

d/dx(sin(x)) = d/dx(g(f(x))) = g'(f(x)) * f'(x)

Let’s calculate the derivatives separately:

f(x) = sin(x)
The derivative of sin(x) with respect to x is cos(x), so f'(x) = cos(x).

g(x) = x
The derivative of x with respect to x is 1, so g'(x) = 1.

Now, substitute these values into the chain rule:

d/dx(sin(x)) = g'(f(x)) * f'(x) = 1 * cos(x) = cos(x)

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

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