Alternative solution approach as introduced in the video Graphs of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math», «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨» «/mi»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi»),«/mi»«/math» and Differentiability.


Starting from the bottom left of the original function, the slopes of the lines tangent to the curve are positive. Moving up and to the right, the slopes remain positive, but the curve is less steep, so the slope decreases. This continues until the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis, where the slope of the line tangent to the curve is zero. Moving up and to the right some more, the slopes are positive again and continue to increase as the curve becomes more steep.



The derivative of a function does not always exist. If the derived function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» is defined at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/mrow»«/mstyle»«/math», that is if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»h«/mi»«/mfrac»«/math» exists, the original function is said to be differentiable at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/mrow»«/mstyle»«/math».

Recall from Unit 1, for a limit to exist, the related one-sided limits must exist and they must be equal. For example, for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math» to exist, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«msup»«mn»0«/mn»«mo»§#8722;«/mo»«/msup»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»h«/mi»«/mfrac»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«msup»«mn»0«/mn»«mo»+«/mo»«/msup»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math».

A function may be differentiable over an interval or it may be differentiable over the entire set of real numbers.