How do you differentiate #e^((x^2-1)^2) # using the chain rule?

Answer 1

#d/dx e^((x^2-1)^2) = 4e^((x^2-1)^2)*x(x^2-1)#

#y = e^((x^2-1)^2)#
Let #u = (x^2-1)^2# such that #y = e^u#
Now #dy/dx = dy/(du) * (du)/(dx)#
To get the derivative of #u#, we use the chain rule again:
Let #v = x^2-1# such that #u = v^2#
#(dv)/(dx) = 2x#
#(du)/(dv) = 2v#
#(du)/(dx) = (dv)/(dx) * (du)/(dv) = 4xv = 4x(x^2-1)#
#dy/(du) = e^u = e^((x^2-1)^2)#

Now we can go back to the original equation:

#dy/dx = dy/(du) * (du)/(dx) = e^((x^2-1)^2) * 4x(x^2-1)#
In the answers, variables #v# and #u# were defined for the use of chain rule.
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Answer 2

To differentiate the function (e^{(x^2-1)^2}) using the chain rule, follow these steps:

  1. Identify the outer function (f(u) = e^u) and the inner function (u(x) = (x^2 - 1)^2).
  2. Find the derivative of the outer function with respect to its variable (u), which is (f'(u) = e^u).
  3. Find the derivative of the inner function with respect to (x), denoted as (u'(x)).
  4. Apply the chain rule, which states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.
  5. Substitute the derivatives of (f(u)) and (u(x)) into the chain rule formula.
  6. Simplify the expression to get the final result.

The derivative of (e^{(x^2-1)^2}) with respect to (x) using the chain rule is:

[ \frac{d}{dx} \left( e^{(x^2-1)^2} \right) = e^{(x^2-1)^2} \cdot 2(x^2-1) \cdot 2x ]

Therefore, the result is (2x \cdot 2(x^2 - 1) \cdot e^{(x^2 - 1)^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.

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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