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What is the second derivative of #f(x)=e^x/x^2 #?

Answer 1

Charles Arocha

Using the quotient rule we get

#f'(x)=[e^x*(x-2)]/x^2#

Using the quotient rule again for the above we get

#f''(x)=[e^x*(x^2-4x+6)]/x^4#

Footnote

The quotient rule for a function such as #f(x)=(p(x))/(q(x))# is

#f'(x)=[p'(x)*q(x)-p(x)*q'(x)]/[q(x)]^2#

where #f'(x)=df(x)/dx#

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Answer 2

Evelyn Agard

To find the second derivative of the function ( f(x) = \frac{e^x}{x^2} ), follow these steps:

Find the first derivative of ( f(x) ) with respect to ( x ).
Once you have the first derivative, differentiate it again with respect to ( x ) to find the second derivative.

Let's begin by finding the first derivative:

[ f'(x) = \frac{d}{dx} \left( \frac{e^x}{x^2} \right) ]

Using the quotient rule:

[ f'(x) = \frac{xe^x(2) - e^x(2x)}{x^4} ]

Simplify:

[ f'(x) = \frac{2xe^x - 2xe^x}{x^3} ]

[ f'(x) = \frac{0}{x^3} ]

[ f'(x) = 0 ]

Now, differentiate ( f'(x) ) to find the second derivative:

[ f''(x) = \frac{d}{dx} (0) ]

[ f''(x) = 0 ]

Therefore, the second derivative of ( f(x) = \frac{e^x}{x^2} ) is ( 0 ).

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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.