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Derivative of a Function











































Derivative using the limit formula

Plug in f(x) = 1/√x into the difference quotient f(x + h) - f(x) / h. Then plug in h = 0 into the final answer. I get 0/2√x, which is just 0. Is this correct?

Comments for Derivative of a Function

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May 16, 2013
Derivative
by: Staff


Answer

Part I


The final answer is not 0.

Since h = 0, the difference quotient f(x + h) - f(x) / h is undefined (you cannot divide by zero).

However, as you have already pointed out, the derivative (or slope) of the function f(x) can be computed using the limit of the difference quotient as h → 0.

f(x + h) - f(x) / h is the slope of a line passing through the two points (x, f(x)) and (x + h, f(x + h)).

The slope at point (x, f(x)) is the limit of the difference quotient as h → 0.

Derivative (slope) at point (x, f(x)) is the limit of the difference quotient as h → 0.





Substituting the function f(x) = 1/√x into the formula, it becomes:

Substituting the function f(x) = 1/√x into the difference quotient limit formula





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May 16, 2013
Derivative
by: Staff


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Part II


To avoid dividing by 0 (when h = 0), the difference quotient just needs to be rearranged algebraically.


Rationalize the numerator by multiplying by its conjugate

The difference quotient needs to be rearranged algebraically.  Rationalize the numerator by multiplying by its conjugate.





Rationalize the numerator by multiplying by its conjugate:  Part I.





Rationalize the numerator by multiplying by its conjugate:  Part II.










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May 16, 2013
Derivative
by: Staff


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Part III


Rationalize the numerator by multiplying by its conjugate:  Part III.





Factor out the h in the denominator, and then cancel with the h in the numerator.

Factor out the h in the denominator, and then cancel with the h in the numerator.





The rearranged (equivalent) fraction avoids the possibility of dividing by zero when h = 0.


The rearranged (equivalent) fraction avoids the possibility of dividing by zero when h = 0.







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May 16, 2013
Derivative
by: Staff


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Part IV


Replace h with 0, and then simplify the expression.

Replace h with 0, and then simplify the expression.





Algebra - Complete the simplification process.






Final Answer:

Derivative using limit formula - final answer.









Thanks for writing.

Staff
www.solving-math-problems.com



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