How do you use the quotient rule to differentiate #3/(3x+3)#?
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To use the quotient rule to differentiate the function (f(x) = \frac{3}{3x+3}), you first need to identify the numerator function (u(x)) and the denominator function (v(x)), then apply the quotient rule formula:
[ f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} ]
For the given function, (u(x) = 3) and (v(x) = 3x + 3).
First, differentiate both (u(x)) and (v(x)):
- (u'(x) = 0) (since the derivative of a constant is zero),
- (v'(x) = 3) (the derivative of (3x + 3) with respect to (x)).
Now, substitute (u, u', v,) and (v') into the quotient rule formula:
[ f'(x) = \frac{(3x + 3) \cdot 0 - 3 \cdot 3}{(3x + 3)^2} = \frac{-9}{(3x + 3)^2} ]
So, the derivative of (\frac{3}{3x+3}) with respect to (x) is (\frac{-9}{(3x + 3)^2}).
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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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