Write down the function for the curve whose tangent line you need to find. Determine at which point you want to take the tangent line (e.g., x = 1).

Take the derivative of the function using the derivative rules. There are too many to summarize here; you can find a list of the rules of derivation under the Resources section, however, in case you need a refresher:

Example: If the function is f(x) = 6x^3 + 10x^2 - 2x + 12, the derivative would be as follows:

f'(x) = 18x^2 + 20x - 2

Note that we represent the derivative of the original function by adding the ' mark, so that f'(x) is the derivative of f(x).

Plug the x-value for which you need the tangent line into f'(x) and calculate what f'(x) will be at that point.

Example: If f'(x) is 18x^2 + 20x - 2 and you need the derivative at the point where x = 0, then you would plug 0 into this equation in place of x to obtain the following:

f'(0) = 18 (0)^2 + 20(0) - 2

so f'(0) = -2.

Write out an equation of the form y = mx + b. This will be your tangent line. m is the slope of your tangent line and it's equal to your result from step 3. You don't know b yet, however, and will need to solve for it. Continuing the example, your initial equation based on step 3 would be y = -2x + b.

Plug the x-value you used to find the slope of the tangent line back into your original equation, f(x). This way, you can determine the y-value of your original equation at this point, then use it to solve for b in your tangent line equation.

Example: If x is 0, and f(x) = 6x^3 + 10x^2 - 2x + 12, then f(0) = 6(0)^3 + 10(0)^2 - 2(0) + 12. All terms in this equation go to 0 except for the last one, so f(0) = 12.

Substitute the result from step 5 for y in your tangent line equation, then substitute the x-value you used in step 5 for x in your tangent line equation and solve for b.

Example: You know from a prior step that y = -2x + b. If y = 12 when x = 0, then 12 = -2(0) + b. The only possible value for b that will give a valid result is 12, therefore b = 12.

Write out your tangent line equation, using the m and b values you have found.

Example: You know m = -2 and b = 12, so y = -2x + 12.