sin^-1(x) derivative
The derivative of sin^-1(x), also written as arcsin(x) or asin(x), is obtained using calculus
The derivative of sin^-1(x), also written as arcsin(x) or asin(x), is obtained using calculus. Let’s go through the steps to find the derivative of sin^-1(x).
Step 1: Start with the definition of arcsin(x).
The definition of arcsin(x) implies that sin(arcsin(x)) = x. In other words, sin^-1(x) is the angle whose sine is x. Rearrange this definition to solve for sin(arcsin(x)) as x.
Step 2: Apply the derivative to both sides.
Now, we will take the derivative of both sides with respect to x. Remembering that sin(arcsin(x)) equals x, we have:
d/dx(sin(arcsin(x))) = d/dx(x).
Step 3: Use the chain rule.
The left-hand side of the equation involves the composition of two functions: sin and arcsin. We apply the chain rule to differentiate this composition. Let u = arcsin(x), so that sin(u) = x. Thus, the chain rule states:
d(sin(u))/du * du/dx = 1.
Step 4: Solve for the derivative of sin(arcsin(x)).
From Step 3, we find that d(sin(u))/du = cos(u), and du/dx is the derivative of arcsin(x). Rewriting this result:
cos(arcsin(x)) * d(arcsin(x))/dx = 1.
Step 5: Solve for d(arcsin(x))/dx.
To isolate d(arcsin(x))/dx, divide both sides of the equation by cos(arcsin(x)):
d(arcsin(x))/dx = 1 / cos(arcsin(x)).
Step 6: Replace cos(arcsin(x)) with a trigonometric identity.
By making use of the right triangle definition of sine and cosine, recall that cos(arcsin(x)) is the square root of 1 minus the square of sin(arcsin(x)). Since sin(arcsin(x)) equals x, we have:
cos(arcsin(x)) = sqrt(1 – x^2).
Step 7: Substitute the expression for cos(arcsin(x)) into d(arcsin(x))/dx.
Replacing cos(arcsin(x)) with sqrt(1 – x^2) in Step 6, we get:
d(arcsin(x))/dx = 1 / sqrt(1 – x^2).
So, the derivative of sin^-1(x) or arcsin(x) with respect to x is 1 / sqrt(1 – x^2).
It is important to mention that the domain for this derivative is -1 ≤ x ≤ 1, as arcsin(x) is only defined within this interval.
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