How do you differentiate #f(x)=8e^(x^2)/(e^x+1)# using the chain rule?

Answer 1

The only trick here is that #(e^(x^2))'=e^(x^2)*(x^2)'=e^(x^2)*2x#
Final derivative is:

#f'(x)=8e^(x^2)(2x*(e^x+1)-e^x)/(e^x+1)^2#
or
#f'(x)=8e^(x^2)(e^x*(2x-1)+2x+1)/(e^x+1)^2#

#f(x)=8(e^(x^2))/(e^x+1)#
#f'(x)=8((e^(x^2))'(e^x+1)-e^(x^2)(e^x+1)')/(e^x+1)^2#
#f'(x)=8(e^(x^2)*(x^2)'(e^x+1)-e^(x^2)*e^x)/(e^x+1)^2#
#f'(x)=8(e^(x^2)2x*(e^x+1)-e^(x^2)*e^x)/(e^x+1)^2#
#f'(x)=8(e^(x^2)(2x*(e^x+1)-e^x))/(e^x+1)^2#
#f'(x)=8e^(x^2)(2x*(e^x+1)-e^x)/(e^x+1)^2#
or (if you want to factor #e^x# in the nominator)
#f'(x)=8e^(x^2)(e^x*(2x-1)+2x+1)/(e^x+1)^2#

Note: if you want to study the sign, you are gonna have a bad time. Just look at the graph:

graph{8(e^(x^2))/(e^x+1) [-50.25, 53.75, -2.3, 49.76]}

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

To differentiate ( f(x) = \frac{8e^{x^2}}{e^x + 1} ) using the chain rule, follow these steps:

  1. Identify the outer function ( u ) and the inner function ( v ).
  2. Compute ( u' ) and ( v' ).
  3. Apply the chain rule: ( \frac{d}{dx}(u(v(x))) = u'(v(x)) \cdot v'(x) ).
  4. Substitute ( u' ), ( v' ), and ( v(x) ) into the chain rule formula.
  5. Simplify the expression.

Here's the step-by-step process:

  1. Let ( u(x) = \frac{8e^x}{e^x + 1} ) and ( v(x) = x^2 ).
  2. Compute ( u'(x) = \frac{8(e^x)(e^x + 1) - 8e^x(e^x)}{(e^x + 1)^2} ) and ( v'(x) = 2x ).
  3. Apply the chain rule: ( \frac{d}{dx}(u(v(x))) = u'(v(x)) \cdot v'(x) ).
  4. Substitute ( u' ), ( v' ), and ( v(x) ): ( \frac{d}{dx}\left(\frac{8e^{x^2}}{e^x + 1}\right) = \frac{8(e^{x^2})(2x)}{e^{x^2} + 1} ).
  5. Simplify the expression: ( \frac{16xe^{x^2}}{e^{x^2} + 1} ).

Therefore, ( \frac{d}{dx}\left(\frac{8e^{x^2}}{e^x + 1}\right) = \frac{16xe^{x^2}}{e^{x^2} + 1} ).

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