Derivative of cos x
The derivative of cos(x) can be found using the basic rules of differentiation
The derivative of cos(x) can be found using the basic rules of differentiation.
The derivative of cos(x) is equal to the negative sine function, denoted as -sin(x).
To understand why the derivative of cos(x) is -sin(x), we can use the definition of the derivative, which states that the derivative of a function is the rate of change of that function with respect to its variable (in this case, x).
The cosine function (cos(x)) represents the ratio of the adjacent side to the hypotenuse in a right triangle, with the angle x being the reference angle. As x increases, the value of cos(x) decreases.
Now, let’s consider the unit circle. The angle x is given by the position of a point on the unit circle. As we move along the circumference of the unit circle counterclockwise, x increases, and the value of cos(x) decreases. On the other hand, the derivative represents the slope of the function at any given point.
For the cosine function, the slope is negative when x is increasing. This means that as x increases, the value of cos(x) decreases, giving us a negative rate of change. Hence, the derivative of cos(x) is -sin(x).
The derivative of cos(x) can also be derived using the trigonometric identity, which states that sin^2(x) + cos^2(x) = 1.
Differentiating both sides with respect to x will give us 2sin(x)cos(x) + 2cos(x)(-sin(x)) = 0.
Simplifying this equation gives us sin(x)cos(x) – cos(x)sin(x) = 0.
Therefore, sin(x)cos(x) = cos(x)sin(x).
Dividing both sides by cos(x), we get sin(x) = 1.
This means that the derivative of cos(x) is -sin(x).
In summary, the derivative of cos(x) is -sin(x).
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