How do you find the equation of the tangent line to the curve?

Find the equation for the tangent line to a curve by finding the derivative of the equation for the curve, then using that equation to find the slope of the tangent line at a given point. Finding the equation for the tangent line requires a knowledge of calculus and the formula for the slope.

1. Find the derivative of the curve equation

   Using techniques of differentiation, find the derivative of the equation for the curve f(x), which is notated as f'(x). The technique needed to differentiate depends on the equation, although common methods include using the chain rule, the product rule and the quotient rule.

2. Find the slope of the tangent line

   After finding the derivative f'(x), plug in a value for x for the point on the curve for which you are calculating the tangent line. The value of f'(x) at that point is the slope of the tangent line.

3. Find the equation of the tangent line in point-slope form

   Express the tangent line equation in point-slope form, which can be found through the equation y1 - y2 = f'(x)(x1 - x2). Using the same point on the line used to find the slope, plug in the coordinates for x1 and y1. Manipulate the equation to express it as y = mx + b.