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Problem 27 Hard Difficulty

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ g(x) = \sqrt{9 - x} $

Answer

$\frac{-1}{2 \sqrt{9-x}}$

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Video Transcript

given a function G of X which is equal to the squared of nine minus x. We are defined the derivative G prime of X by definition of derivative and the domain of G F X and G prime of X. Now, by limit definition we have G prime of X. This is equal to The limit as h approaches zero of G of X plus H minus tree of X. All over H. And so we have a limit as h approaches zero of The square root of nine minus exports each minus the square root of nine minus X. This all over H. Now from here we want an rationalize the numerator and so we multiply this by the conjugate Square it of 9- Express age Plus the square root of 9 -1 over the same expression square it of nine minus x. S H Plus the Square Root of nine -X. Now simplifying this, we get limit as h approaches zero of the square of the square it of 9 -1 was each minus the square of the square root of nine minus x. This all over H times we have squared up nine minus expose age Plus the Square Root of nine -X. And so simplifying further we get limit has asia approaches zero of we have nine minus export h minus. We have nine minus x over h times the square root of nine minus x plus H Plus the Square Root of nine -X. And so we get limit. As a joke approaches zero of nine minus x minus H -9 Plus X. This all over each times the square root of nine minus experts, H Plus the Square Root of nine -X. And so combining like terms we get limit as h approaches zero of you have negative age over age times the square root of nine minus expose each Plus the Square Root of nine -X. From here, we can reduce this to limit as H approaches zero of negative one over the square root of nine minus x psh Plus the Square Root of nine -X. And evaluating at H equals zero. We get negative one over The square root of 9 -60 Plus The Spirit of nine -X. This will give us -1 over squared of nine minus x plus the square root of nine minus x, which is just negative one over Two times the square root of nine -X. Therefore this is the derivative of the function. How far the domain of G and G prime If G of X, this is equal to the square root of nine -X. Then We say that 9 -1 must be Greater than or equal to zero. And so solving for X, we get nine greater than or equal to X. And so the domain of the function G is equal to Negative infinity to nine including 9. Now, if the derivative G prime of X, this is equal to negative one over Two times the square root of nine -X. Then, since the denominator cannot be zero, then we have nine -X must be Greater than zero only. And solving for X, we get nine greater than X and so the domain of the derivative is Negative Infinity to nine, excluding nine.