Understanding the Derivative of the Cosine Function | Explained Step-by-Step

(d/dx) cos(x)

The expression (d/dx) cos(x) represents the derivative of the cosine function with respect to x

The expression (d/dx) cos(x) represents the derivative of the cosine function with respect to x. To find the derivative, we can apply the rules of differentiation.

The derivative of the cosine function is given by:

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

This means that the rate of change of the cosine function with respect to x is equal to the negative sine of x. In other words, as x increases, the value of cos(x) decreases at a rate equal to the sine of x.

To understand why the derivative of cos(x) is -sin(x), we can look at the definition of the derivative. The derivative of a function is defined as the limit of the difference quotient as the interval between two points approaches zero. In the case of cos(x), we can write the difference quotient as:

(d/dx) cos(x) = lim(h->0) [cos(x + h) – cos(x)] / h

By applying trigonometric identities and simplifying the expression, we can get:

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

So, the derivative of cos(x) is -sin(x).

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