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Differentiate f(t) = √(36 ln(t) + 5t)

f'(t) = (1/2) * (36 ln(t) + 5t)^-1/2 * (36/t + 5)

f'(t) = (1/2)* (36/t + 5)/((36 ln(t) + 5t)^1/2

f'(t) = (36 + 5t) / (2t √(36lnt + 5t))

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Find the largest & smallest values of:

F(x) 10e^(g(x)), g(x) = x² - 24x for 0 ≤ x ≤ 13

g'(x) = 2x - 24
0 = 2x - 24
x = 12

g(12) = -144
g(0) = 0
g(13) = -143

Largest value: 10
Smallest value: e^(-144)

***Smallest value: 10e^(-144)

Agree with the first one

Agree with the second one

Differentiate f(t) = √(36 ln(t) + 5t):

To differentiate f(t), we will use the chain rule. The chain rule states that if we have a composition of functions, we need to differentiate the outer function and multiply it by the derivative of the inner function.

In this case, the outer function is the square root, and the inner function is 36 ln(t) + 5t.

Step 1: Find the derivative of the outer function.
To find the derivative of the square root function, we use the power rule, which states that if we have a function raised to a power of 1/2, we bring down the power and multiply by the derivative of the function inside the square root.

The derivative of √u, where u is a function, is (1/2) * u^(-1/2).

Step 2: Find the derivative of the inner function.
To find the derivative of 36 ln(t) + 5t, we need to use the sum rule.
The sum rule states that if we have a function that is a sum of two or more functions, we can find the derivative of each term separately and add them together.

The derivative of 36 ln(t) is (36/t) by the logarithmic differentiation rule.
The derivative of 5t is 5.

Step 3: Combine the derivatives.
Multiplying the derivative of the outer function from Step 1 by the derivative of the inner function from Step 2, we get:

f'(t) = (1/2) * (36 ln(t) + 5t)^(-1/2) * ((36/t) + 5)

Simplifying the expression, we can write:

f'(t) = (1/2) * (36/t + 5)/√(36 ln(t) + 5t)

Therefore, the correct derivative is f'(t) = (1/2) * (36/t + 5)/√(36 ln(t) + 5t).

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For the second part of the question, finding the largest and smallest values of F(x) = 10e^(g(x)), where g(x) = x² - 24x for 0 ≤ x ≤ 13:

Step 1: Find the critical points by finding where the derivative of g(x) is equal to zero or does not exist.
The derivative of g(x) is g'(x) = 2x - 24.

Setting g'(x) = 0, we have:
2x - 24 = 0
2x = 24
x = 12

So x = 12 is a critical point.

Step 2: Evaluate g(x) at the critical point and the endpoints of the given interval:
g(12) = (12)^2 - 24(12) = -144
g(0) = (0)^2 - 24(0) = 0
g(13) = (13)^2 - 24(13) = -143

Step 3: Evaluate F(x) at the critical point and the endpoints of the given interval:
F(x) = 10e^(g(x))

Largest value: Find the maximum value of e^(g(x)) from the critical point and the endpoints. Since e^x is an increasing function, the largest value of e^(g(x)) will be at g(0) = 0.

Therefore, the largest value of F(x) = 10e^(g(x)) is F(0) = 10e^0 = 10.

Smallest value: Find the minimum value of e^(g(x)) from the critical point and the endpoints. Since e^x is always positive, the minimum value occurs at g(12) = -144.

Therefore, the smallest value of F(x) = 10e^(g(x)) is F(12) = 10e^(-144).

Hence, the correct largest value is 10 and the correct smallest value is e^(-144).