# How do you find the equation of the tangent and normal line to the curve #y=9x^-1# at x-3?

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If

Tangent is

If

Tangent is

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First we need the derivative, which will give us the slope of the tangent at any point:

We will use the formula

When x=3

Tangent:

Passes through

Normal:

Passes through

When x=-3

Tangent:

Passes through

Normal:

Passes through

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To find the equation of the tangent line to the curve y = 9x^(-1) at x = 3, we first find the derivative of the function y with respect to x, which is dy/dx = -9x^(-2).

Then, evaluate dy/dx at x = 3 to find the slope of the tangent line:

dy/dx = -9(3)^(-2) = -9/9 = -1.

So, the slope of the tangent line is -1 at x = 3.

Now, we use the point-slope form of a line to find the equation of the tangent line. Given the point (3, 9/3) on the curve, and the slope -1, the equation of the tangent line is:

y - y₁ = m(x - x₁), where (x₁, y₁) = (3, 9/3).

Substitute the values into the equation:

y - 3 = -1(x - 3).

Now, simplify the equation:

y - 3 = -x + 3.

y = -x + 6.

This is the equation of the tangent line.

To find the equation of the normal line, we use the fact that the slope of the normal line is the negative reciprocal of the slope of the tangent line. So, the slope of the normal line is 1.

Using the point-slope form again, with the point (3, 9/3) and slope 1, we get:

y - 3 = 1(x - 3).

Simplify:

y - 3 = x - 3.

y = x.

This is the equation of the normal line.

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To find the equation of the tangent line to the curve ( y = 9x^{-1} ) at ( x = 3 ), you first need to find the derivative of the function with respect to ( x ). Then, evaluate the derivative at ( x = 3 ) to find the slope of the tangent line. Next, use the point-slope form of a line with the point ( (3, 3) ) and the slope you found to write the equation of the tangent line. The normal line will have a slope that is the negative reciprocal of the slope of the tangent line, and you can use the same point to find the equation of the normal line.

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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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