Derivative of f(x) = cos(x) using the chain rule explained step-by-step

d/dx cosx

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

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

The chain rule states that if we have a function composed with another function, the derivative can be found by taking the derivative of the outer function multiplied by the derivative of the inner function.

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

So, we can apply the chain rule as follows:

f'(x) = (d/dx) [ cos(g(x)) ]

Now, let’s find the derivative of the outer function (cos(x)) and the derivative of the inner function (x) separately.

The derivative of cos(x) with respect to x is given by:

(d/dx) [ cos(x) ] = -sin(x)

And the derivative of g(x) = x with respect to x is simply 1.

Now, applying the chain rule:

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

Therefore, the derivative of f(x) = cos(x) with respect to x is f'(x) = -sin(x).

Hope this explanation helps! Let me know if you have any further questions.

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