Derivative of sin(x) with Respect to x: Applying the Chain Rule and Other Methods

d/dx(sinx)

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

To find the derivative of sin(x) with respect to x, we can apply the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) with respect to x is given by the derivative of f evaluated at g(x), multiplied by the derivative of g with respect to x.

In this case, f(u) = sin(u) and g(x) = x. So, we can write sin(x) as f(g(x)).

Applying the chain rule, the derivative of sin(x) with respect to x, d/dx(sin(x)), is given by:

d/dx(sin(x)) = d/du(sin(u)) * d/dx(x)

Now, d/du(sin(u)) is the derivative of sin(u) with respect to u. Since sin(u) is a trigonometric function, its derivative is cos(u):

d/du(sin(u)) = cos(u)

d/dx(x) is the derivative of x with respect to x, which is simply 1:

d/dx(x) = 1

Therefore, we have:

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

Simplifying further, we can substitute u with x:

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

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

Alternatively, we can also differentiate sin(x) using the other derivative rules. The derivative of sin(x) is given by:

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

This result can also be obtained by differentiating the sine function using the limit definition of the derivative. However, using the chain rule is usually the quicker and more straightforward method when differentiating composition of functions.

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