how to find the equation of the line tangent to the function at the given point
find first derivative, plug x value given into first derivative to get slope, put into point slope form using the point given and the slope you found
To find the equation of the line tangent to a function at a given point, follow these steps:
1. Find the derivative of the function using the power rule, product rule, or chain rule, depending on the complexity of the function.
2. Plug in the x-coordinate of the given point into the derivative to find the slope of the tangent line at that point.
3. 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 found in step 2, to write the equation of the tangent line.
4. Simplify the equation by solving for y, if necessary.
Example: Find the equation of the line tangent to the function f(x) = x^2 + 3x + 1 at the point (-2, 1).
1. Find the derivative of the function: f'(x) = 2x + 3.
2. Plug in x = -2 to find the slope of the tangent line: f'(-2) = 2(-2) + 3 = -1.
3. Use the point-slope form of a line: y – 1 = -1(x – (-2)).
4. Simplify the equation by solving for y: y = -x + 3.
Therefore, the equation of the line tangent to f(x) = x^2 + 3x + 1 at (-2, 1) is y = -x + 3.
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