Understanding the Derivative of Sin(x) using the Chain Rule | Cos(x)

Derivative of sin x

The derivative of sin x can be found by using the chain rule and the derivative of the function sin x is cos x

The derivative of sin x can be found by using the chain rule and the derivative of the function sin x is cos x.

To understand this, let’s look at the definition of the derivative of a function.

The derivative of a function f(x) represents the rate at which the function is changing at any point x on its graph. It gives us the slope of the tangent line to the graph of the function at that point.

In the case of the sine function, sin x represents the value of the y-coordinate on the unit circle corresponding to an angle x. When we take the derivative of sin x, we are actually finding the rate at which the y-coordinate changes as x changes.

To find the derivative of sin x, we need to apply the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of that composition is given by the product of the derivative of the outer function (f'(g(x))) and the derivative of the inner function (g'(x)).

For sin x, the outer function is sin and the inner function is x. The derivative of the outer function sin x is cos x. The derivative of the inner function x is 1. Therefore, using the chain rule, we can say that the derivative of sin x is given by:

d/dx(sin x) = cos x * 1 = cos x

So, the derivative of sin x is cos x. This means that the rate at which the y-coordinate on the unit circle changes as the angle x changes is given by the function cos x.

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