How do you use the definition of a derivative to find the derivative of #g(x) = sqrt(9 − x)#?
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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.
- Start with the given function ( g(x) = \sqrt{9 - x} ).
- Apply the definition of the derivative ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
- Substitute ( g(x) ) into the formula to get ( g'(x) = \lim_{h \to 0} \frac{\sqrt{9 - (x + h)} - \sqrt{9 - x}}{h} ).
- Simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is ( \sqrt{9 - (x + h)} + \sqrt{9 - x} ).
- This yields ( g'(x) = \lim_{h \to 0} \frac{9 - (x + h) - (9 - x)}{h(\sqrt{9 - (x + h)} + \sqrt{9 - x})} ).
- Simplify the numerator to get ( g'(x) = \lim_{h \to 0} \frac{-h}{h(\sqrt{9 - (x + h)} + \sqrt{9 - x})} ).
- Cancel out the ( h ) terms to get ( g'(x) = \lim_{h \to 0} \frac{-1}{\sqrt{9 - (x + h)} + \sqrt{9 - x}} ).
- As ( h ) approaches 0, the expression ( \sqrt{9 - (x + h)} ) approaches ( \sqrt{9 - x} ).
- 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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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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