{"id":3009,"date":"2025-08-15T20:03:20","date_gmt":"2025-08-15T20:03:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=3009"},"modified":"2025-08-15T20:29:36","modified_gmt":"2025-08-15T20:29:36","slug":"derivatives-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/derivatives-learn-it-2\/","title":{"raw":"Derivatives: Learn It 2","rendered":"Derivatives: Learn It 2"},"content":{"raw":"

Finding Derivatives of Rational Functions<\/h2>\r\nTo find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.\r\n\r\n
Find the derivative of the function [latex]f\\left(x\\right)=\\frac{3+x}{2-x}[\/latex] at [latex]x=a[\/latex].[reveal-answer q=\"183162\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183162\"]\r\n

[latex]\\begin{align}{f}^{\\prime }\\left(a\\right)&=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h} \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{\\frac{3+\\left(a+h\\right)}{2-\\left(a+h\\right)}-\\left(\\frac{3+a}{2-a}\\right)}{h}&& \\text{Substitute }f\\left(a+h\\right)\\text{ and }f\\left(a\\right) \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)\\left[\\frac{3+\\left(a+h\\right)}{2-\\left(a+h\\right)}-\\left(\\frac{3+a}{2-a}\\right)\\right]}{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)\\left(h\\right)}&& \\text{Multiply numerator and denominator by }\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right) \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{\\bcancel{\\left(2-\\left(a+h\\right)\\right)}\\left(2-a\\right)\\left(\\frac{3+\\left(a+h\\right)}{\\bcancel{\\left(2-\\left(a+h\\right)\\right)}}\\right)-\\left(2-\\left(a+h\\right)\\right)\\bcancel{\\left(2-a\\right)}\\left(\\frac{3+a}{\\bcancel{2-a}}\\right)}{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)\\left(h\\right)}&& \\text{Distribute and cancel} \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{6 - 3a+2a-{a}^{2}+2h-ah - 6+3a+3h - 2a+{a}^{2}+ah}{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)\\left(h\\right)}&& \\text{Multiply} \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{5\\cancel{h}}{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)\\left(\\cancel{h}\\right)}&& \\text{Combine like terms} \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{5}{\\left(2-\\left(a+h\\right)\\right)\\left(2-a\\right)}&& \\text{Cancel like factors} \\\\ &=\\frac{5}{\\left(2-\\left(a+0\\right)\\right)\\left(2-a\\right)}=\\frac{5}{\\left(2-a\\right)\\left(2-a\\right)}=\\frac{5}{{\\left(2-a\\right)}^{2}}&& \\text{Evaluate the limit} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

\r\n
\r\n\r\nFind the derivative of the function [latex]f\\left(x\\right)=\\frac{10x+11}{5x+4}[\/latex] at [latex]x=a[\/latex].\r\n\r\n[reveal-answer q=\"395414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"395414\"]\r\n\r\n[latex]\\begin{align}{f}^{\\prime }\\left(a\\right)=\\frac{-15}{{\\left(5a+4\\right)}^{2}}\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
[ohm_question hide_question_numbers=1]174109[\/ohm_question]<\/section>
\r\n

Finding Derivatives of Functions with Roots<\/h2>\r\nTo find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.\r\n\r\n
Find the derivative<\/strong> of the function [latex]f\\left(x\\right)=4\\sqrt{x}[\/latex] at [latex]x=36[\/latex].[reveal-answer q=\"388460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388460\"]\r\n\r\nWe have\r\n

[latex]\\begin{align}{f}^{\\prime }\\left(a\\right)&=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h} \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\frac{4\\sqrt{a+h}-4\\sqrt{a}}{h}&& \\text{Substitute }f\\left(a+h\\right)\\text{ and }f\\left(a\\right) \\end{align}[\/latex]<\/p>\r\nMultiply the numerator and denominator by the conjugate: [latex]\\frac{4\\sqrt{a+h}+4\\sqrt{a}}{4\\sqrt{a+h}+4\\sqrt{a}}[\/latex].\r\n

[latex]\\begin{align}{f}^{\\prime }\\left(a\\right)&=\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\frac{4\\sqrt{a+h}-4\\sqrt{a}}{h}\\right)\\cdot \\left(\\frac{4\\sqrt{a+h}+4\\sqrt{a}}{4\\sqrt{a+h}+4\\sqrt{a}}\\right) \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\frac{16\\left(a+h\\right)-16a}{h\\left(4\\sqrt{a+h}+4\\sqrt{a}\\right)}\\right)&& \\text{Multiply}. \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\frac{16a+16h-16a}{h\\left(4\\sqrt{a+h}+4\\sqrt{a}\\right)}\\right) && \\text{Distribute and combine like terms}. \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\frac{16\\cancel{h}}{\\cancel{h}\\left(4\\sqrt{a+h}+4\\sqrt{a}\\right)}\\right)&& \\text{Simplify}. \\\\ &=\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\frac{16}{4\\sqrt{a+h}+4\\sqrt{a}}\\right)&& \\text{Evaluate the limit by letting }h=0. \\\\ &=\\frac{16}{8\\sqrt{a}}=\\frac{2}{\\sqrt{a}} \\\\ {f}^{\\prime }\\left(36\\right)&=\\frac{2}{\\sqrt{36}} =\\frac{2}{6} =\\frac{1}{3} && \\text{Evaluate the derivative at }x=36. \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

Find the derivative of the function [latex]f\\left(x\\right)=9\\sqrt{x}[\/latex]\u00a0at [latex]x=9[\/latex].[reveal-answer q=\"854877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"854877\"][latex]\\frac{3}{2}[\/latex][\/hidden-answer]<\/section>
[ohm_question hide_question_numbers=1]174117[\/ohm_question]<\/section><\/section>","rendered":"

Finding Derivatives of Rational Functions<\/h2>\n

To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.<\/p>\n

Find the derivative of the function [latex]f\\left(x\\right)=\\frac{3+x}{2-x}[\/latex] at [latex]x=a[\/latex].<\/p>\n