How I do I prove the Quotient Rule for derivatives?
We can use the product rule:
In order to prove the quotient rule, which states that
Use the product rule on this:
Hence we obtain
Rewriting with fractions, this becomes
Rewriting with a common denominator, this becomes
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To prove the Quotient Rule for derivatives, follow these steps:

Start with the quotient of two functions: ( f(x) = \frac{g(x)}{h(x)} ).

Let ( u(x) = g(x) ) and ( v(x) = \frac{1}{h(x)} ). Rewrite the quotient as ( f(x) = u(x)v(x) ).

Use the Product Rule to find ( f'(x) = u'(x)v(x) + u(x)v'(x) ).

Substitute ( u(x) = g(x) ) and ( v(x) = \frac{1}{h(x)} ) into the expression.

Differentiate ( u(x) ) and ( v(x) ) with respect to ( x ).

Simplify the expression obtained in step 3 using the derivatives of ( u(x) ) and ( v(x) ).

Rewrite the expression in terms of ( g(x) ) and ( h(x) ) to obtain the Quotient Rule for derivatives:
[ \frac{d}{dx}\left(\frac{g(x)}{h(x)}\right) = \frac{g'(x)h(x)  g(x)h'(x)}{(h(x))^2} ]
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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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