Let {{MATH_214}}. Find an equation for the line tangent to the curve {{MATH_215}} at the point {{MATH_216}}. {{CMD_217}} To find an equation for a line, we need two pieces of information. We need to know a point on the line, and we need to know the slope of the line. In this question, we are given that the point {{MATH_218}} is on the line. So, we need to find the slope of the tangent line. Finding the slope of the tangent line at the point {{MATH_219}} means finding {{MATH_220}}. Start by writing out the definition of the derivative, {{MATH_221}} Multiply by {{MATH_222}} to clear the fraction in the numerator, {{MATH_223}} Combine like-terms in the numerator, {{MATH_224}} Take the limit as {{MATH_225}} goes to {{MATH_226}}, {{MATH_227}} We are looking for an equation of the line through the point {{MATH_228}} with slope {{MATH_229}}. The point-slope formula tells us that the line has equation given by {{MATH_230}} or equivalently, {{MATH_231}}. {{CMD_232}} We can confirm our results by looking at the graph of {{MATH_233}} and the line {{MATH_234}}. {{CMD_235}}{{ENV_236}}{{CMD_237}} {{CMD_238}} {{CMD_239}} {{CMD_240}}