The derivative of a function gives the instantaneous rate of change for a given point. Think of the way the velocity of a car is always changing as it accelerates and decelerates. Although you can calculate the average velocity for the entire trip, sometimes you need to know the velocity for a particular instant. The derivative provides this information, not just for velocity but for any rate of change. A tangent line shows what could have been if the rate had been constant, or what could be if it remains unchanged.
Determine the coordinates of the indicated point by plugging the value of x into the function. For example, to find the tangent line where x = 2 of the function F(x) = -x^2 + 3x, plug x into the function to find F(2) = 2. Thus the coordinate would be (2, 2).
Find the derivative of the function. Think of the derivative of a function as a formula that gives the slope of the function for any value of x. For example, the derivative F'(x) = -2x + 3.
Calculate the slope of the tangent line by plugging the value of x into the function of the derivative. For example, slope = F'(2) = -2 * 2 + 3 = -1.
Find the y-intercept of the tangent line by subtracting the slope times the x-coordinate from the y-coordinate: y-intercept = y1 - slope * x1. The coordinate found in Step 1 must satisfy the tangent line equation. Therefore plugging in the coordinate values into the slope-intercept equation for a line, you can solve for the y-intercept. For example, y-intercept = 2 - (-1 * 2) = 4.
Write the equation of the tangent line in the form y = slope * x + y-intercept. In the example given, y = -x + 4.
Pick another point and find the equation of the tangent line for the function given in the example.