How do you find f'(x) using the definition of a derivative for #f(x)=(x6)^(2/3)#?
Short answer: use
In this case we have cube roots of squares, but the idea is the same.
From the identity above, we get:
Using the above algebra, we can find:
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To find ( f'(x) ) using the definition of a derivative for ( f(x) = (x  6)^{2/3} ), you can follow these steps:

Start with the definition of the derivative: [ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h)  f(x)}}{h} ]

Substitute the function ( f(x) ) into the definition: [ f'(x) = \lim_{{h \to 0}} \frac{{((x + h)  6)^{2/3}  (x  6)^{2/3}}{h} ]

Expand the terms: [ f'(x) = \lim_{{h \to 0}} \frac{{((x + h)  6)^{2/3}  (x  6)^{2/3}}{h} ]

Apply the binomial expansion formula: [ (a + b)^{2/3} = a^{2/3} + \frac{2}{3}a^{1/3}b + \frac{1}{3}a^{2/3}b^2 ]

Substitute the expressions into the expanded form: [ f'(x) = \lim_{{h \to 0}} \frac{{((x  6) + h)^{2/3}  (x  6)^{2/3}}{h} ]

Apply the binomial expansion formula to each term: [ ((x  6) + h)^{2/3} = (x  6)^{2/3} + \frac{2}{3}(x  6)^{1/3}h + \frac{1}{3}(x  6)^{2/3}h^2 ]

Substitute the expressions into the derivative formula: [ f'(x) = \lim_{{h \to 0}} \frac{{(x  6)^{2/3} + \frac{2}{3}(x  6)^{1/3}h + \frac{1}{3}(x  6)^{2/3}h^2  (x  6)^{2/3}}{h} ]

Simplify and cancel terms: [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{2}{3}(x  6)^{1/3}h + \frac{1}{3}(x  6)^{2/3}h^2}}{h} ] [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{2}{3}h + \frac{1}{3}(x  6)^{1/3}h^2}}{h} ]

Cancel the ( h ) terms: [ f'(x) = \lim_{{h \to 0}} \frac{{\frac{2}{3} + \frac{1}{3}(x  6)^{1/3}h}}{1} ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = \frac{2}{3} ]
So, the derivative ( f'(x) ) of ( f(x) = (x  6)^{2/3} ) is ( \frac{2}{3} ).
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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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