Derivative of cos x
The derivative of the cosine function, cos(x), can be found using the rules of calculus
The derivative of the cosine function, cos(x), can be found using the rules of calculus.
The derivative of a function represents the rate of change of the function at any given point. In the case of cos(x), the derivative will give us the rate at which the cosine function is changing with respect to the variable x.
The derivative of the cosine function is equal to the negative sine function, -sin(x). In mathematical notation, this can be written as:
d/dx(cos(x)) = -sin(x)
To understand why this is the case, we can use the definition of the derivative. The derivative of a function f(x) with respect to x is defined as the limit of the difference quotient as the change in x approaches zero:
f'(x) = lim(h -> 0) [(f(x + h) – f(x))/h]
Applying this definition to the cosine function, we have:
d/dx(cos(x)) = lim(h -> 0) [(cos(x + h) – cos(x))/h]
Using the trigonometric identity for the difference of cosines, we can rewrite this expression as:
d/dx(cos(x)) = lim(h -> 0) [-2sin((x + h + x)/2)sin((x + h – x)/2)/h]
Simplifying further, we have:
d/dx(cos(x)) = lim(h -> 0) [-2sin((2x + h)/2)sin(h/2)/h]
As h approaches zero, sin(h/2)/h tends to 1, so we can replace it in the expression:
d/dx(cos(x)) = -2sin(x)
Therefore, the derivative of cos(x) is -sin(x).
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