# How do you find the slope & equation of tangent line to #f(x)=-1/x# at (3,-1/3)?

Slope

Given function:

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To find the slope of the tangent line to the function f(x) = -1/x at the point (3, -1/3), we need to find the derivative of the function and evaluate it at x = 3.

The derivative of f(x) = -1/x can be found using the quotient rule. Applying the quotient rule, we have:

f'(x) = (0 * x - (-1) * 1) / (x^2) = 1/x^2

To find the slope of the tangent line at x = 3, we substitute x = 3 into the derivative:

f'(3) = 1/(3^2) = 1/9

Therefore, the slope of the tangent line to f(x) = -1/x at (3, -1/3) is 1/9.

To find the equation of the tangent line, we use the point-slope form of a linear equation:

y - y1 = m(x - x1)

Substituting the values of the point (3, -1/3) and the slope 1/9, we have:

y - (-1/3) = (1/9)(x - 3)

Simplifying the equation, we get:

y + 1/3 = (1/9)(x - 3)

Multiplying through by 9 to eliminate the fraction, we have:

9y + 3 = x - 3

Rearranging the equation to the standard form, we get:

x - 9y = 6

Therefore, the equation of the tangent line to f(x) = -1/x at (3, -1/3) is x - 9y = 6.

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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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