How do you differentiate #f(x)=2^(-x^2)#?

Answer 1

Please see below.

For #a > 0#, #d/dx(a^x) = a^x lna#
For exponent other then #x#, we use the chain rule, so that
#d/dx(a^u) = a^u ln(a) (du)/dx#

So for this question, we have

#f'(x) = 2^(-x^2) ln(2) * (-2x)#
# = -2x2^(-x^2)ln(2)#
Or #-x2^(-x^2+1)ln(2)#
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Answer 2

To differentiate the function f(x) = 2^(-x^2), you would use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

The derivative of 2^(-x^2) with respect to x is -2x * 2^(-x^2) * ln(2). Therefore, the derivative of f(x) = 2^(-x^2) is f'(x) = -2x * 2^(-x^2) * ln(2).

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Answer 3

To differentiate ( f(x) = 2^{-x^2} ), we can use the chain rule of differentiation. The chain rule states that if we have a function of the form ( g(h(x)) ), then its derivative is ( g'(h(x)) \cdot h'(x) ).

For ( f(x) = 2^{-x^2} ), we can rewrite it as ( f(x) = (2^{x})^{-x} ). Now, we can apply the chain rule.

Let ( g(x) = 2^x ) and ( h(x) = -x^2 ).

Then, ( f(x) = g(h(x)) ).

Using the chain rule, the derivative of ( f(x) ) with respect to ( x ) is:

[ f'(x) = g'(h(x)) \cdot h'(x) ]

[ = (2^x \cdot \ln(2)) \cdot (-2x) ]

[ = -2x \cdot 2^x \cdot \ln(2) ]

So, the derivative of ( f(x) = 2^{-x^2} ) is ( f'(x) = -2x \cdot 2^x \cdot \ln(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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