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How do you differentiate #f(x)=1/x+1/x^3# using the sum rule?

Answer 1

Christopher Allers

#f'(x)=-(x^2+3)/x^4#

Find the derivative of each individual part using the power rule:

#d/dx[x^n]=nx^(n-1)#

Thus,

#f'(x)=d/dx[x^-1]+d/dx[x^-3]#

#f'(x)=-x^-2-3x^-4#

#f'(x)=-1/x^2-3/x^4#

This can also be written as:

#f'(x)=-(x^2+3)/x^4#

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Answer 2

Aurora Alvarado

To differentiate $f(x) = \frac{1}{x} + \frac{1}{x^3}$ using the sum rule, you differentiate each term separately and then sum the results.

So, for the first term, $\frac{1}{x}$ , the derivative is $-\frac{1}{x^2}$ .

For the second term, $\frac{1}{x^3}$ , the derivative is $-\frac{3}{x^4}$ .

Adding these derivatives together, you get:

$f'(x) = -\frac{1}{x^2} - \frac{3}{x^4}$ .

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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.