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How do you differentiate #f(x)=3/x-5/(1-x)# using the sum rule?

Answer 1

Isabella Ackerman

#color(maroon)(f’(x) = -(3/x^2 + 5/(1-x)^2)#

#f(x) = (3/x) - (5/(1-x))#

#f(x) = 3x^-1 - 5(1-x)^-1#

#f’(x) = (3 * -1* x^-2) - (5 * -1 (1-x)^-2 * -1)#

#f’(x) = -3x^-2 - 5(1-x)^-2#

#color(maroon)(f’(x) = -(3/x^2 + 5/(1-x)^2)#

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Answer 2

Axel Alvarez

To differentiate ( f(x) = \frac{3}{x} - \frac{5}{1-x} ) using the sum rule, you differentiate each term separately and then sum the results.

( f(x) = \frac{3}{x} - \frac{5}{1-x} )

First term: ( \frac{3}{x} ) Using the power rule, the derivative is ( -\frac{3}{x^2} ).

Second term: ( -\frac{5}{1-x} ) Using the chain rule, the derivative is ( -\frac{5}{(1-x)^2} ).

Summing the derivatives of each term, we get:

( f'(x) = -\frac{3}{x^2} - \frac{5}{(1-x)^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.