How do you use the definition of a derivative to find the derivative of #g(x) = sqrt(9 − x)#?

Answer 1

#g'(x) = 1/(2sqrt(9-x))#

From first principles, #g'(x) = Lim_"h->0" (g(x+h) - g(x))/h#
In this example #g(x) = sqrt(9-x)#
Therefore #g'(x) = Lim_"h->0" (sqrt(9-(x+h)) -sqrt(9-x))/h#
Multiply top and bottom by #(sqrt(9-(x+h)) +sqrt(9-x)) ->#
#g'(x) = Lim_"h->0" (9-(x+h) -(9-x))/ (h (sqrt(9-(x+h)) +sqrt(9-x))#
# =Lim_"h->0" h/ (h (sqrt(9-(x+h)) +sqrt(9-x))#
# =Lim_"h->0" cancel h/ (cancel h (sqrt(9-(x+h)) +sqrt(9-x))#
#= 1/ ( (sqrt(9-(x+0)) +sqrt(9-x))#
#=1/(2sqrt(9-x))#
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Answer 2

To find the derivative of ( g(x) = \sqrt{9 - x} ), we can use the definition of a derivative, which states that the derivative of a function is the limit of the difference quotient as the interval approaches zero.

  1. Start with the given function ( g(x) = \sqrt{9 - x} ).
  2. Apply the definition of the derivative ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
  3. Substitute ( g(x) ) into the formula to get ( g'(x) = \lim_{h \to 0} \frac{\sqrt{9 - (x + h)} - \sqrt{9 - x}}{h} ).
  4. Simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is ( \sqrt{9 - (x + h)} + \sqrt{9 - x} ).
  5. This yields ( g'(x) = \lim_{h \to 0} \frac{9 - (x + h) - (9 - x)}{h(\sqrt{9 - (x + h)} + \sqrt{9 - x})} ).
  6. Simplify the numerator to get ( g'(x) = \lim_{h \to 0} \frac{-h}{h(\sqrt{9 - (x + h)} + \sqrt{9 - x})} ).
  7. Cancel out the ( h ) terms to get ( g'(x) = \lim_{h \to 0} \frac{-1}{\sqrt{9 - (x + h)} + \sqrt{9 - x}} ).
  8. As ( h ) approaches 0, the expression ( \sqrt{9 - (x + h)} ) approaches ( \sqrt{9 - x} ).
  9. Thus, the limit becomes ( g'(x) = \frac{-1}{2\sqrt{9 - x}} ).

Therefore, the derivative of ( g(x) = \sqrt{9 - x} ) is ( g'(x) = \frac{-1}{2\sqrt{9 - x}} ).

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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.

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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