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How do you find the derivative of #arcsin(x^2)#?

Answer 1

Christopher Allers

# d/dx arcsin(x^2)= (2x)/sqrt(1 -x^4) #

Let # y = arcsin(x^2) => siny = x^2 #

Differentiating wrt x;

# cosydy/dx = 2x #

Using the identity # sin^2A+cos^2A -= 1 #

# sin^2y + cos^2y = 1 # # :. (x^2)^2 + cos^2y = 1 # # :. cos^2y = 1 -x^4 # # :. cosy = sqrt(1 -x^4) #

And so;

# sqrt(1 -x^4)dy/dx = 2x # # :. dy/dx = (2x)/sqrt(1 -x^4) #

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Answer 2

Luke Alvarez

To find the derivative of arcsin(x^2), you can use the chain rule. Let y = arcsin(x^2).

Find the derivative of y with respect to x: dy/dx = d(arcsin(u))/du * du/dx, where u = x^2.
Compute the derivatives:
- d(arcsin(u))/du = 1 / sqrt(1 - u^2)
- du/dx = 2x
Substitute the derivatives back into the chain rule formula.
Simplify the expression to get the final derivative.

Therefore, the derivative of arcsin(x^2) with respect to x is dy/dx = (2x) / sqrt(1 - (x^2)^2).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.