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Using the limit definition, how do you differentiate #f(x) =9-x^2#?

Answer 1

The derivative is defined as the limit of the incremental ratio, so:

#f'(x) = lim_(Delta x ->0) (f(x+Delta x) - f(x))/(Delta x)#

#(Delta f(x))/(Delta x) = (9 - (x+Delta x)^2 - 9 + x^2)/(Delta x)#

#(Delta f(x))/(Delta x) = (cancel(9) cancel (- x^2) -2xDelta x-(Delta x)^2cancel(- 9) + cancel(x^2))/(Delta x) =-2x- Delta x#

Carrying to the limit:

#lim_(Delta x ->0) (Delta f(x))/(Delta x) = -2x#

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Answer 2

Elijah Abram

To differentiate the function ( f(x) = 9 - x^2 ) using the limit definition of a derivative, we use the formula:

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

First, we plug the given function into the formula:

[ f'(x) = \lim_{h \to 0} \frac{(9 - (x + h)^2) - (9 - x^2)}{h} ]

Then, we expand and simplify:

[ f'(x) = \lim_{h \to 0} \frac{9 - (x^2 + 2xh + h^2) - 9 + x^2}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-x^2 - 2xh - h^2 + x^2}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{-2xh - h^2}{h} ]

[ f'(x) = \lim_{h \to 0} -2x - h ]

[ f'(x) = -2x ]

So, the derivative of ( f(x) = 9 - x^2 ) with respect to ( x ) is ( f'(x) = -2x ).

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Answer from HIX Tutor

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.