# How do you find the equation of a line tangent to the function #y=x^2(x-2)^3# at x=1?

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The equation is

The first step is to find the derivative. This will give us important information about the slope of the tangent. We will use the chain rule to find the derivative.

Finally, we can construct the equation of the tangent.

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To find the equation of a line tangent to a function at a specific point, you can follow these steps:

- Find the derivative of the function.
- Substitute the given x-value into the derivative to find the slope of the tangent line.
- Use the point-slope form of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope, to write the equation of the tangent line.

For the function y = x^2(x - 2)^3, the derivative is y' = 6x(x - 2)^2 + 2x^2(x - 2)^3.

Substituting x = 1 into the derivative, we get y' = 6(1)(1 - 2)^2 + 2(1)^2(1 - 2)^3 = -6.

The slope of the tangent line at x = 1 is -6.

Using the point-slope form with the given point (1, f(1)), we can write the equation of the tangent line as y - f(1) = -6(x - 1).

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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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