MIT grad shows how to find the equation of a tangent line using derivatives (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For a more complex example that uses the CHAIN RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS, skip to time 11:35.
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What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus to find the tangent line by taking the derivative of the given equation. Here are the steps to finding the equation of the tangent line, using derivatives:
1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x.
2) PLUG IN X-VALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given x-value into the derivative of f(x). The value you get is the slope of the tangent line, m.
3) FIND Y-VALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the y-value of the point. You get this by plugging the given x-value into the original equation to find the corresponding y-value. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a y-value of 7. This means that the full point is (4,7).
4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINT-SLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point-slope equation of a line. The point-slope equation is x - x1 = m (y - y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.
You can also rearrange the equation you get so that it is in "y equals" or slope-intercept form, if you want.
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