# How do you use the limit definition to find the derivative of #x/(sqrt(1-x^2)#?

Substitute equation [1] and [2] into equation [3]:

Simplify the numerator (a lot):

Simplify the denominator:

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To find the derivative of ( \frac{x}{\sqrt{1-x^2}} ) using the limit definition of the derivative, we first express the function in a form suitable for this method.

The limit definition of the derivative of a function ( f(x) ) is given by:

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

In this case, let ( f(x) = \frac{x}{\sqrt{1-x^2}} ). To find ( f'(x) ), we need to evaluate the limit as ( h \to 0 ) of the expression ( \frac{f(x + h) - f(x)}{h} ).

First, find ( f(x + h) ):

[ f(x + h) = \frac{x + h}{\sqrt{1 - (x + h)^2}} ]

Next, find ( f(x) ):

[ f(x) = \frac{x}{\sqrt{1 - x^2}} ]

Now, subtract ( f(x) ) from ( f(x + h) ):

[ f(x + h) - f(x) = \frac{x + h}{\sqrt{1 - (x + h)^2}} - \frac{x}{\sqrt{1 - x^2}} ]

Simplify the expression:

[ f(x + h) - f(x) = \frac{(x + h)\sqrt{1 - x^2} - x\sqrt{1 - (x + h)^2}}{\sqrt{1 - x^2}\sqrt{1 - (x + h)^2}} ]

Now, divide by ( h ) and take the limit as ( h \to 0 ) to find the derivative. This process involves some algebraic manipulation and can be complex, so it's often easier to use trigonometric substitution or other methods to simplify the expression before taking the limit.

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