Learn How To Find The Derivative Of Arcsin(X) With Step-By-Step Instructions

d arcsinx

1/sqrt(1-x^2)

The derivative of arcsin(x) is:

d/dx arcsin(x) = 1 / sqrt(1 – x^2)

To derive this result, we will use the chain rule and the fact that the derivative of sin(x) is cos(x). First, we write arcsin(x) as a function of sin(x):

arcsin(x) = sin^-1(x)

Next, we use the chain rule:

d/dx arcsin(x) = d/dx sin^-1(x) = d/dx sin(y), where y = sin^-1(x)

Now, we can use the formula for the derivative of sin(x):

d/dx sin(y) = cos(y) * dy/dx

We just need to solve for dy/dx. To do this, we use the fact that sin(y) = x, so we take the sine of both sides:

sin(y) = x
==> y = sin^-1(x)

Thus, we have:

dy/dx = cos(y) = cos(sin^-1(x))

Now, we substitute this back in to our original expression:

d/dx arcsin(x) = cos(sin^-1(x)) * dy/dx

d/dx arcsin(x) = cos(sin^-1(x)) * cos(sin^-1(x))

Recall that sin^2(x) + cos^2(x) = 1. If we substitute x = sin(y) into this expression, we get:

sin^2(y) + cos^2(y) = 1

==> cos^2(y) = 1 – sin^2(y)

==> cos(y) = sqrt(1 – sin^2(y))

Thus:

d/dx arcsin(x) = sqrt(1 – sin^2(sin^-1(x)))

d/dx arcsin(x) = sqrt(1 – x^2) / 1 [since sin(arcsin(x)) = x]

Therefore, the derivative of arcsin(x) is 1 / sqrt(1 – x^2).

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