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

The slope of the tangent at x = a on f(x) is f'(a)

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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) = x^2/(x+1), you can use the quotient rule. The quotient rule states that if you have a function in the form f(x) = g(x)/h(x), then the derivative of f(x) is given by [g'(x)h(x) - g(x)h'(x)] / [h(x)]^2.

Applying the quotient rule to f(x) = x^2/(x+1), we have:

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

Simplifying this expression, we get:

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

Combining like terms, we have:

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

Now, to find the slope of the tangent line at the point (2, 4/3), we substitute x = 2 into the derivative expression:

f'(2) = (2^2 + 2(2)) / (2^2 + 2(2) + 1)

Simplifying this expression, we get:

f'(2) = (4 + 4) / (4 + 4 + 1)

f'(2) = 8 / 9

Therefore, the slope of the tangent line to the graph of f(x) = x^2/(x+1) at the point (2, 4/3) is 8/9.

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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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