There are several ways in which you can find the slope of a tangent to a function. These include actually drawing a plot of the function and the tangent line and physically measuring the slope and also using successive approximations via secants. However, for simple algebraic functions, the quickest approach is to use calculus. The calculus method takes the derivative of the function at the point of interest, which is equal to the slope of the tangent at that point.
Write out the equation of the function to which you are going to apply a tangent. It should be written in the form of y = f(x). As an example, consider the function y = 4x^3 + 2x - 6.
Take the first derivative of this function. To take the derivative, rewrite each term of the function, changing terms of the form ax^b to (a)(b)x^(b-1). When rewriting terms, note that x^0 has a value of 1. Also, terms in the initial function that are purely numerical are dropped entirely when writing the derivative. So, for the example function, the first derivative would be y'(x) = 12x^2 + 2. The "tick" mark after the y shows this is a derivative.
Determine the x value of the point on the function where you want the tangent line located. Insert this value into the derivative wherever x occurs. In the example, if you wanted to find the tangent to the function at the point with x = 3, you would write y'(3) = 12(3^2) + 2.
Solve for the function with the value for x you just inserted. The example function is 12(9) + 2 = 110. This is the slope of the tangent line to the original function at that x value.
Because the tangent line will be horizontal at a maximum or minimum point of a curved function, it will have a slope of zero. This fact is sometimes used to find maxima and minima of functions, because their first derivative will be zero at those points.