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

Answer 1

Luke Alvarez

#( 2x)/(sqrt(1-x^4) #

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

( this should be known as it is fairly standard )

for the question here the derivative is the same but # x = x^2#

differentiate using the #color(blue)(" chain rule ") #

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

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

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Answer 2

Luke Alvarez

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

Let y = arcsin(x^2). Then, take the derivative of both sides with respect to x:

dy/dx = d(arcsin(x^2))/dx

Apply the chain rule:

dy/dx = (1/sqrt(1 - (x^2)^2)) * d(x^2)/dx

Now, find d(x^2)/dx, which is 2x.

Substitute d(x^2)/dx = 2x into the expression:

dy/dx = (1/sqrt(1 - (x^2)^2)) * 2x

Simplify:

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

So, the derivative of arcsin(x^2) with respect to x is 2x / sqrt(1 - x^4).

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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.