What is the point-slope form of the equation of the tangent line to the curve y = 5x² - 10x + 10 at the point x = 3?

To determine the point-slope form of the equation of the tangent line to the curve ( y = 5x^2 - 10x + 10 ) at the point where ( x = 3 ), we first need to find both the slope of the tangent line at that point and the coordinates of the point on the curve.

The slope of the tangent line can be found by taking the derivative of the function, which gives us the slope of the tangent at any point ( x ).

1. Find the derivative:

[

\frac{dy}{dx} = 10x - 10

]

2. Evaluate the derivative at ( x = 3 ):

[

\frac{dy}{dx} \bigg|_{x=3} = 10(3) - 10 = 30 - 10 = 20

]

So, the slope of the tangent line at ( x = 3 ) is 20.

3. Find the coordinates of the point on the curve at ( x = 3 ):

[

y = 5(3)^2 - 10(3