Find the derivative of the function using the definition of derivative.

g(t)= 9/sqrt(t)
g'(t)=

state the domain of the function and the domain of the derivative (use interval notation)

i have the g'(t) it is -9/2t^3/2

whats the domain?

To find the derivative of the function g(t) = 9/sqrt(t) using the definition of derivative, we need to apply the limit definition:

g'(t) = lim (h->0) [g(t+h) - g(t)] / h

Let's go through the steps:

Step 1: Replace g(t) with its original expression:
g(t) = 9/sqrt(t)

Step 2: Replace g(t+h) with its original expression:
g(t+h) = 9/sqrt(t + h)

Step 3: Substitute the expressions into the definition of the derivative:
g'(t) = lim (h->0) [9/sqrt(t + h) - 9/sqrt(t)] / h

Step 4: Rationalize the numerator using the conjugate:
g'(t) = lim (h->0) [(9/sqrt(t + h) - 9/sqrt(t)) * (√(t + h) + √t)] / h * (√(t + h) + √t)

Step 5: Simplify the numerator by distributing and combining like terms:
g'(t) = lim (h->0) [9(√(t + h) + √t) - 9(√(t + h) - √t)] / h * (√(t + h) + √t)

Step 6: Simplify further:
g'(t) = lim (h->0) [9√(t + h) + 9√t - 9√(t + h) + 9√t] / h * (√(t + h) + √t)

Step 7: Cancel out the common terms in the numerator:
g'(t) = lim (h->0) [18√t] / h * (√(t + h) + √t)

Step 8: Simplify the expression:
g'(t) = lim (h->0) (18√t) / h * (2√t + √(t + h))

Step 9: Divide the numerator by h:
g'(t) = lim (h->0) 18 / (2√t + √(t + h))

Now we can simplify the expression further if needed, but this is the derivative of the function g(t) using the definition of derivative.

The domain of the original function g(t) = 9/sqrt(t) is t > 0 since we have √(t) in the denominator.

The domain of the derivative g'(t) is the same as the domain of the original function, t > 0, because we didn't introduce any new restrictions or undefined values.

To find the derivative of the function g(t) = 9/sqrt(t) using the definition of derivative, we need to use the limit definition:

g'(t) = lim(h→0) [g(t + h) - g(t)] / h

Let's begin by plugging in the given function into the derivative definition:

g'(t) = lim(h→0) [9/sqrt(t + h) - 9/sqrt(t)] / h

Next, let's simplify the expression by finding a common denominator:

g'(t) = lim(h→0) [9*sqrt(t) - 9*sqrt(t + h)] / (h*sqrt(t)*sqrt(t + h))

Now, let's factor out a common factor of 9 and combine the two terms:

g'(t) = lim(h→0) [9 * (sqrt(t) - sqrt(t + h))] / (h*sqrt(t)*sqrt(t + h))

To simplify the expression further, we can rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is sqrt(t) + sqrt(t + h):

g'(t) = lim(h→0) (9 * (sqrt(t) - sqrt(t + h))) * (sqrt(t) + sqrt(t + h)) / (h*sqrt(t)*sqrt(t + h) * (sqrt(t) + sqrt(t + h)))

Now, let's simplify the expression:

g'(t) = lim(h→0) [9 * (t - (t + h))] / (h*sqrt(t)*sqrt(t + h) * (sqrt(t) + sqrt(t + h)))

g'(t) = lim(h→0) [9 * (t - t - h)] / (h*sqrt(t)*sqrt(t + h) * (sqrt(t) + sqrt(t + h)))

g'(t) = lim(h→0) [-9h] / (h*sqrt(t)*sqrt(t + h) * (sqrt(t) + sqrt(t + h)))

Now, we can cancel out the h from the numerator and denominator:

g'(t) = lim(h→0) -9 / (sqrt(t)*sqrt(t + h) * (sqrt(t) + sqrt(t + h)))

Finally, we can take the limit as h approaches 0:

g'(t) = -9 / (sqrt(t)^2 * (sqrt(t) + sqrt(t)))

g'(t) = -9 / (t * (2*sqrt(t)))

g'(t) = -9 / (2t*sqrt(t))

Therefore, the derivative of the function g(t) = 9/sqrt(t) using the definition of derivative is g'(t) = -9 / (2t*sqrt(t)).

Now, let's determine the domain of the function g(t) and its derivative g'(t):

The function g(t) = 9/sqrt(t) is defined for all positive values of t since the square root of a negative number is undefined.

Domain of g(t): t > 0

For the derivative g'(t) = -9 / (2t*sqrt(t)), the denominator 2t*sqrt(t) will be defined as long as t > 0.

Domain of g'(t): t > 0

In interval notation, the domain of both g(t) and g'(t) is (0, ∞).