How do you differentiate #f(x)=x/(x4)^2# using the quotient rule?
See below.
The quotient rule is given by
In this case,
As indicated, we will first take the derivative of
From this, we subtract the product of
And,
The derivative is then given as:
Which simplifies to
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To differentiate ( f(x) = \frac{x}{(x4)^2} ) using the quotient rule:
 Identify ( u(x) = x ) and ( v(x) = (x  4)^2 ).
 Apply the quotient rule formula:
[ f'(x) = \frac{u'(x)v(x)  u(x)v'(x)}{[v(x)]^2} ]

Calculate the derivatives of ( u(x) ) and ( v(x) ):
 ( u'(x) = 1 ) (derivative of ( x ))
 ( v'(x) = 2(x  4) ) (derivative of ( (x  4)^2 ))

Plug the derivatives and functions into the quotient rule formula:
[ f'(x) = \frac{(1)(x  4)^2  (x)(2(x  4))}{(x  4)^4} ]
 Simplify the expression:
[ f'(x) = \frac{(x  4)^2  2x(x  4)}{(x  4)^4} ]
 Expand and combine like terms:
[ f'(x) = \frac{x^2  8x + 16  2x^2 + 8x}{(x  4)^4} ]
[ f'(x) = \frac{x^2 + 16}{(x  4)^4} ]
Thus, the derivative of ( f(x) ) is ( f'(x) = \frac{x^2 + 16}{(x  4)^4} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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