How do you differentiate # f(x)= (x^2 + 2x )/( x^2 -4)#?

Answer 1

#f'(x)=-2/(x-2)^2#

Since the given function is in the form of a fraction, we can use the quotient rule.

The quotient rule states that,

#color(blue)(|bar(ul(color(white)(a/a)color(black)((u/v)'=(vu'-uv')/(v^2))color(white)(a/a)|)))#

In your case:

Let #x^2+2x# be #u#. Let #x^2-4# be #v#.

Plugging in the values into the quotient rule,

#f'(x)=((x^2-4)(x^2+2x)'-(x^2+2x)(x^2-4)')/(x^2-4)^2#
Use the power rule to differentiate #(x^2+2x)# and #(x^2-4)#.
Note: Use the formula, #(x^n)'=n*x^(n-1)#.
#f'(x)=((x^2-4)(2x+2)-(x^2+2x)(2x))/(x^2-4)^2#

Simplifying,

#f'(x)=((2x^3+2x^2-8x-8)-(2x^3+4x^2))/((x+2)^2(x-2)^2)#

Removing the brackets in the numerator,

#f'(x)=(2x^3+2x^2-8x-8-2x^3-4x^2)/((x+2)^2(x-2)^2)#
#f'(x)=(color(red)cancelcolor(black)(2x^3)+2x^2-8x-8color(red)cancelcolor(black)(-2x^3)-4x^2)/((x+2)^2(x-2)^2)#
#f'(x)=(-2x^2-8x-8)/((x+2)^2(x-2)^2)#
Factor out #-2# from the numerator.
#f'(x)=(-2(x^2+4x+4))/((x+2)^2(x-2)^2)#

Factor the leftover expression in the numerator.

#f'(x)=(-2(x+2)^2)/((x+2)^2(x-2)^2)#
#f'(x)=(-2color(red)cancelcolor(black)((x+2)^2))/(color(red)cancelcolor(black)((x+2)^2)(x-2)^2)#
#f'(x)=color(green)(|bar(ul(color(white)(a/a)color(black)(-2/(x-2)^2)color(white)(a/a)|)))#
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Answer 2

To differentiate the function ( f(x) = \frac{x^2 + 2x}{x^2 - 4} ), you would use the quotient rule, which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Applying the quotient rule, you would first find the derivatives of the numerator and the denominator, and then substitute them into the formula to find the derivative of the function. The derivatives of the numerator and denominator are ( 2x + 2 ) and ( 2x ), respectively. Substituting these into the quotient rule formula, the derivative of ( f(x) ) would be ( \frac{(2x + 2)(x^2 - 4) - (x^2 + 2x)(2x)}{(x^2 - 4)^2} ). Simplifying this expression yields the final answer for the derivative of ( f(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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