Since the derivative gives us a formula for the slope of a tangent line to a curve, we can gain information about a function purely from the sign of the derivative. In particular, we have the following theorem {{CMD_102}}{{CMD_103}} A function {{MATH_104}} is {{CMD_105}}increasing{{CMD_106}} on any interval {{MATH_107}} where {{MATH_108}}, for all {{MATH_109}} in {{MATH_110}}.{{CMD_111}} A function {{MATH_112}} is {{CMD_113}}decreasing{{CMD_114}} on any interval {{MATH_115}} where {{MATH_116}}, for all {{MATH_117}} in {{MATH_118}}.{{CMD_119}} {{CMD_120}}