# How do you find the slope of a tangent line to the graph of the function #f(x)= (1+2x^(1/2)) / (1+x^(3/2))# at (4, 5/9)?

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To find the slope of a tangent line to the graph of a function at a specific point, you can use the derivative of the function.

To find the derivative of f(x), you can use the quotient rule.

The derivative of f(x) = (1+2x^(1/2)) / (1+x^(3/2)) is given by:

f'(x) = [(1+x^(3/2))(2(1/2)x^(-1/2)) - (1+2x^(1/2))(3/2)x^(1/2)] / (1+x^(3/2))^2

Simplifying this expression, we get:

f'(x) = (2x^(-1/2) + 3x^(1/2)) / (2(1+x^(3/2))^2)

To find the slope of the tangent line at (4, 5/9), substitute x = 4 into the derivative expression:

f'(4) = (2(4)^(-1/2) + 3(4)^(1/2)) / (2(1+(4)^(3/2))^2)

Simplifying this expression, we get:

f'(4) = (1/4 + 3/2) / (2(1+8))^2

f'(4) = (7/4) / (18)^2

f'(4) = 7/1296

Therefore, the slope of the tangent line to the graph of f(x) = (1+2x^(1/2)) / (1+x^(3/2)) at (4, 5/9) is 7/1296.

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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.

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