Let {{MATH_166}} be a function given by {{MATH_167}}. What is the instantaneous rate of change of {{MATH_168}} at {{MATH_169}}, where {{MATH_170}} is any real number? {{CMD_171}} The rate of change is given by {{MATH_172}}, and so is the slope of the tangent line to the curve {{MATH_173}} at the point {{MATH_174}}. {{CMD_175}} {{CMD_176}} The line tangent to the curve {{MATH_177}}, is simply the curve itself! {{CMD_178}} {{CMD_179}} The derivative is the slope of the line {{MATH_180}}! Therefore, {{MATH_181}}, for all real numbers {{MATH_182}}. {{CMD_183}} {{CMD_184}} We can verify this reasoning by computing {{MATH_185}}, using the definition of the derivative. {{MATH_186}} Replace {{MATH_187}} with its formula, {{MATH_188}} Simplify the numerator, {{MATH_189}} Factor the numerator, {{MATH_190}} Compute the limit, {{MATH_191}} {{CMD_192}} {{CMD_193}}