How do you differentiate #arctan(x^2)#?

Answer 1

#(2x)/(1+x^4)#

The derivative of the arctangent function is:

#d/dxarctan(x)=1/(1+x^2)#

So, when applying the chain rule, this becomes

#d/dxarctan(f(x))=1/(1+(f(x))^2)*f'(x)#
So, for #arctan(x^2)#, where #f(x)=x^2#, we have
#d/dxarctan(x^2)=1/(1+(x^2)^2)*d/dx(x^2)#
#=(2x)/(1+x^4)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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

Let #y=arctan(x^2)#.
Then #tan(y)=x^2#. From here, differentiate both sides of the equation. Recall that the chain rule comes into effect on the left-hand side.
#sec^2(y)*dy/dx=2x#
Dividing both sides by #sec^2(y)#, which is equivalent to multiplying both sides by #cos^2(y)#, gives
#dy/dx=cos^2(y)*2x#
#dy/dx=cos^2(arctan(x^2))*2x#
Note that #cos^2(arctan(x^2))# can be simplified.
If #arctan(x^2)# is an angle in a right triangle, then #x^2# is the side opposite the angle and #1# is the side adjacent to the angle. Then, by the Pythagorean Theorem, #sqrt(1+x^4)# is the triangle's hypotenuse.
Since cosine is the adjacent side, #1#, divided by the hypotenuse, #sqrt(1+x^4)#, we see that #cos(arctan(x^2))=1/sqrt(1+x^4)#.

Therefore

#dy/dx=(1/sqrt(1+x^4))^2*2x=1/(1+x^4)*2x=(2x)/(1+x^4)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To differentiate arctan(x^2), you can use the chain rule. Let u = x^2, then differentiate u with respect to x, du/dx = 2x. Now differentiate arctan(u) with respect to u, d(arctan(u))/du = 1/(1+u^2). By the chain rule, the derivative of arctan(u) with respect to x is the product of these two derivatives: (1/(1+u^2)) * (2x). Substituting u = x^2 back in, the derivative is 2x/(1 + x^4).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7