How do you differentiate #f(x)=sqrt(e^(-x^2+x) # using the chain rule?

Answer 1

#(df)/(dx)=(1-2x)/2sqrt(e^(-x^2+x))#

We have to use here the concept of function of a function which uses the formula of chain rule.

As #f(x)=sqrt(g(x))# where #g(x)=e^(h(x))# and #h(x)==-x^2+x# according to chain rule
#(df)/(dx)=(df)/(dg)xx(dg)/(dx)xx(dh)/(dx)#
Hence #(df)/(dx)=1/(2sqrt(e^(-x^2+x)))xxe^(-x^2+x)xx(-2x+1)#
or #(df)/(dx)=(1-2x)/2sqrt(e^(-x^2+x))#
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Answer 2

To differentiate the function ( f(x) = \sqrt{e^{-x^2 + x}} ) using the chain rule, you first identify the outer function as ( g(u) = \sqrt{u} ) and the inner function as ( u = e^{-x^2 + x} ). Then, you find the derivative of the outer function with respect to its variable ( u ), which is ( g'(u) = \frac{1}{2\sqrt{u}} ). Next, you find the derivative of the inner function with respect to ( x ), which is ( u'(x) = \frac{d}{dx} e^{-x^2 + x} ). Finally, you apply the chain rule, which states that the derivative of the composition of functions is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. Therefore, the derivative of ( f(x) ) with respect to ( x ) is ( f'(x) = g'(u) \cdot u'(x) = \frac{1}{2\sqrt{u}} \cdot \frac{d}{dx} e^{-x^2 + x} ).

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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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