d/dx arccos(x)
To find the derivative of the function f(x) = arccos(x) with respect to x, we can use the chain rule
To find the derivative of the function f(x) = arccos(x) with respect to x, we can use the chain rule.
Recall that the derivative of the inverse trigonometric functions can be derived using the basic trigonometric functions. Specifically, the derivative of arccos(x) with respect to x is given by:
d/dx arccos(x) = -1 / √(1 – x^2).
To understand how this derivative is derived, let’s go step by step:
1. Start with the equation: cos(y) = x.
Take the arccos on both sides: arccos(cos(y)) = arccos(x).
Simplify to: y = arccos(x).
2. Now, differentiate both sides of the equation (with respect to x):
d/dx(y) = d/dx(arccos(x)).
The left-hand side of the equation is simply dy/dx, which represents the derivative of y with respect to x.
3. Differentiate the right-hand side using the chain rule:
d/dx(arccos(x)) = d/dx(y).
We need to find dy/dx.
4. Recall the trigonometric identity: cos^2(x) + sin^2(x) = 1.
Differentiate both sides: 2cos(x)(-sin(x)) + 2sin(x)cos(x) = 0.
Simplify: 2cos(x)(-sin(x) + sin(x)cos(x)) = 0.
Divide both sides by cos^2(x): -sin(x) + sin(x)cos(x) = 0.
5. Rearrange the equation: sin(x)cos(x) = sin(x).
Divide both sides by sin(x): cos(x) = 1.
6. Substitute cos(x) = 1 in the equation y = arccos(x) obtained in step 1:
y = arccos(1).
Since the range of arccos(x) is [0, π], the only possible value for y is 0.
7. Hence, dy/dx = 0.
Therefore, d/dx arccos(x) = 0.
It’s important to note that the derivative of arccos(x) is a constant function, which means it does not depend on the value of x.
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