How do I find an equation of the tangent line to the curve y = x³ + 2x² at the point (2, 6)?

To find the equation of the tangent line to the curve at a given point, we need to follow these steps:

1. Find the derivative of the function: The first step is to compute the derivative of the function, which gives us the slope of the tangent line at any point on the curve.
2. For the function y = x³ + 2x², we apply the power rule:
• The derivative, y’ = 3x² + 4x.
3. Evaluate the derivative at the given point: Next, we substitute the x-coordinate of the point (which is 2) into the derivative to find the slope at that point.
• y’(2) = 3(2)² + 4(2) = 3(4) + 8 = 12 + 8 = 20.
4. Use the point-slope form of the line: With the slope (20) and the point (2, 6), we can use the point-slope form of a line, which is given by y – y₁ = m(x – x₁), where m is the slope and (x₁, y₁) is the point.
• Here, substituting the values: y – 6 = 20(x – 2).
5. Simplify to find the equation: Finally, we can simplify this equation.
• y – 6 = 20x – 40
• y = 20x – 34

Thus, the equation of the tangent line to the curve at the point (2, 6) is:

y = 20x – 34