How do you differentiate #arcsin(csc(x^3)) )# using the chain rule?

Answer 1

#(dy)/dx=(3x^2csc(x^3)cot(x^3))/sqrt(1-(csc(x^3))^2)#

Given: #y=arcsin(csc(x^3))# #arcsin(csc(x^3))=sin^-1(csc(x^3))#
Let #t=csc(x^3)# #y=sin^-1t# #(dy)/dt=1/sqrt(1-t^2)=1/sqrt(1-(csc(x^3))^2#
#(dy)/dx=(dy)/(dt)(dt)/dx#
#t=csc(x^3)#
Let #u=x^3#
#t=cscu#
#(dt)/(du)=cscucotu=csc(x^3)cot(x^3)#
#(dt)/dx=(dt)/(du)(du)/(dx)#
#(du)/(dx)=3x^2#
#(dt)/dx=(csc(x^3)cot(x^3))(3x^2)=3x^2csc(x^3)cot(x^3)#
#(dy)/dx=1/sqrt(1-(csc(x^3))^2)(3x^2csc(x^3)cot(x^3))#
#(dy)/dx=(3x^2csc(x^3)cot(x^3))/sqrt(1-(csc(x^3))^2)#
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Answer 2

To differentiate arcsin(csc(x^3)) using the chain rule, follow these steps:

  1. Start with the outer function, arcsin(u), where u = csc(x^3).
  2. Find the derivative of the outer function: d(arcsin(u))/du = 1/sqrt(1 - u^2).
  3. Now, apply the chain rule by finding the derivative of the inner function.
  4. The inner function is u = csc(x^3), so its derivative is du/dx = -3x^2 * csc(x^3) * cot(x^3).
  5. Finally, multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative: = (1/sqrt(1 - u^2)) * (-3x^2 * csc(x^3) * cot(x^3)).
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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.

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.

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.

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.

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