How do you differentiate #f(x) =(1+2x)/(-x^2+1)# using the quotient rule?

Answer 1

The quotient rule states that for #f(x)=g(x)/h(x)#, then #(df(x))/(dx)=(g'(x)h(x)-g(x)h'(x))/h(x)^2#

Derivating it following the quotient rule, then:

#(df(x))/(dx)=(2(-x^2+1)-(1+2x)(-2x))/(-x^2+1)^2#
#(df(x))/(dx)=(-2x^2+2+2x+4x^2)/(1-x^2)^2#
#(df(x))/(dx)=(2(x^2+x+1))/((1-x^2)^2#
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Answer 2

To differentiate ( f(x) = \frac{1 + 2x}{-x^2 + 1} ) using the quotient rule:

  1. Identify ( u = 1 + 2x ) and ( v = -x^2 + 1 ).
  2. Apply the quotient rule formula: ( f'(x) = \frac{v(u') - u(v')}{v^2} ).
  3. Calculate ( u' ) and ( v' ).
  4. Substitute the values into the quotient rule formula and simplify to get the derivative ( f'(x) ).
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Answer 3

To differentiate the function ( f(x) = \frac{1 + 2x}{-x^2 + 1} ) using the quotient rule:

  1. Identify ( u(x) ) as the numerator and ( v(x) ) as the denominator.
  2. Apply the quotient rule: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ).
  3. Find ( u'(x) ) and ( v'(x) ).
  4. Substitute these values into the quotient rule formula.
  5. Simplify the expression to get the derivative.
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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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