The tangent to a curve is a straight line that touches the curve at a certain point and has exactly the same slope as the curve at that point. There will be a different tangent for each point of a curve, but by using calculus you will be able to calculate the tangent line to any point of a curve if you know the function that generates the curve. In calculus, the derivative of a function is the slope of the function at a certain point, and so the tangent line to the curve.
Write down the equation of the function that defines the curve, in the form y = f(x). For example, use y = x^2 + 3.
Rewrite each term of the function, changing each term of the form ax^b to a_b_x^(b-1). If a term has no x value, remove it from the rewritten function. This is the derivative function of the original curve. For the example function, the calculated derivative function f'(x) is f'(x) = 2*x.
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Find the value on the horizontal axis or x value of the point of the curve you want to calculate the tangent for and replace x on the derivative function by that value. To calculate the tangent of the example function at the point where x = 2, the resulting value would be f'(2) = 2*2 = 4. This is the slope of the tangent to the curve at that point.
Calculate the function for the tangent line using the equation for a straight line -- f(x) = a*x + c. Replace a with the calculated tangent slope and c with the value of any term on the original function that had no x values. In the example, the tangent line equation of y = x^2 + 3 at the point where x = 2 would be y = 4x + 3.
Draw the tangent line to the curve if required. Calculate the value of the tangent function for a second value of x such as x + 1 and draw a line between the tangent point and the second calculated point. Using the example, calculate y for x=3 obtaining y = 4*3 + 3 = 15. The straight line that passes the points (11, 2) and (15, 3) is the mathematical tangent to the curve.