# How do you find the derivative by definition for #y=x^(7/3)#?

The details depend on whether you use

or,

Either way, we'll start with the same algebra:

So we can 'rationalize' differences involving cube roots:

This may be easier to see written:

This allows us to rewrite the fraction with a numerator that is a difference of 7th powers.

Then, depending on the form of the definition you are using:

Or, expand (using the binomial theorem)

Finally, reduce the fraction and evaluate the limit.

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To find the derivative of ( y = x^{\frac{7}{3}} ) by definition, we use the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

Substitute ( f(x) = x^{\frac{7}{3}} ) into the definition:

[ f'(x) = \lim_{h \to 0} \frac{(x+h)^{\frac{7}{3}} - x^{\frac{7}{3}}}{h} ]

Expand ( (x+h)^{\frac{7}{3}} ) using binomial theorem:

[ f'(x) = \lim_{h \to 0} \frac{(x^{\frac{7}{3}} + \frac{7}{3}x^{\frac{4}{3}}h + \frac{21}{9}x^{\frac{1}{3}}h^2 + \frac{35}{27}h^3) - x^{\frac{7}{3}}}{h} ]

Simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{\frac{7}{3}x^{\frac{4}{3}}h + \frac{21}{9}x^{\frac{1}{3}}h^2 + \frac{35}{27}h^3}{h} ]

[ f'(x) = \lim_{h \to 0} \left( \frac{7}{3}x^{\frac{4}{3}} + \frac{21}{9}x^{\frac{1}{3}}h + \frac{35}{27}h^2 \right) ]

Now, as ( h \to 0 ), the term ( \frac{21}{9}x^{\frac{1}{3}}h ) and ( \frac{35}{27}h^2 ) approach 0:

[ f'(x) = \frac{7}{3}x^{\frac{4}{3}} ]

Therefore, the derivative of ( y = x^{\frac{7}{3}} ) by definition is ( \frac{7}{3}x^{\frac{4}{3}} ).

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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.

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