$\phantom{\rule{2.77695pt}{0ex}}{f}^{\prime }\left(1\right)\phantom{\rule{2.77695pt}{0ex}}$ for the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{2}+6x-5$ ]]> $8$ ]]>$3$ ]]>$2$ ]]>$\frac{10}{3}$ ]]>$-3$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=cosx-sinx$ ]]> ${f}^{\prime }\left(x\right)=sinx-cosx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ ]]>${f}^{\prime }\left(x\right)=sinx+cosx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ ]]>${f}^{\prime }\left(x\right)=cosx-sinx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ ]]>${f}^{\prime }\left(x\right)=-cosx-sinx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ ]]>${f}^{\prime }\left(x\right)=-1-tanx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=\sqrt{x}$ ]]> ${f}^{\prime }\left(x\right)=\frac{2}{\sqrt{x}}$ ]]>${f}^{\prime }\left(x\right)=-\frac{2}{\sqrt{x}}$ ]]>${f}^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$ ]]>${f}^{\prime }\left(x\right)=-\frac{1}{2\sqrt{x}}$ ]]>${f}^{\prime }\left(x\right)=\frac{2}{3}{x}^{3∕2}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={e}^{3x}$ ]]> ${f}^{\prime }\left(x\right)={e}^{3x}$ ]]>${f}^{\prime }\left(x\right)=3x{e}^{3x}$ ]]>${f}^{\prime }\left(x\right)=\frac{{e}^{3x}}{3}$ ]]>${f}^{\prime }\left(x\right)={e}^{3x}+3$ ]]>${f}^{\prime }\left(x\right)=3{e}^{3x}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{2}lnx$ ]]> ${f}^{\prime }\left(x\right)=x$ ]]>${f}^{\prime }\left(x\right)=2x+\frac{1}{x}$ ]]>${f}^{\prime }\left(x\right)=2x-\frac{1}{x}$ ]]>${f}^{\prime }\left(x\right)=x\left(2lnx+1\right)$ ]]>${f}^{\prime }\left(x\right)=2xlnx+\frac{1}{x}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=\frac{x-1}{{x}^{2}+1}$ ]]> ${f}^{\prime }\left(x\right)=\frac{1+2x-{x}^{2}}{{\left({x}^{2}+1\right)}^{2}}$ ]]>${f}^{\prime }\left(x\right)=\frac{1+2x-{x}^{2}}{{x}^{2}+1}$ ]]>${f}^{\prime }\left(x\right)=\frac{{\left(x+1\right)}^{2}}{{x}^{2}+1}$ ]]>${f}^{\prime }\left(x\right)=\frac{{\left(x-1\right)}^{2}}{{x}^{2}+1}$ ]]>${f}^{\prime }\left(x\right)=\frac{1-2x-{x}^{2}}{{\left({x}^{2}+1\right)}^{2}}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}y={x}^{3}-x+4\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ at the point $\phantom{\rule{2.77695pt}{0ex}}\left(-1,4\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$. ]]> $y=-4x$ ]]>$y=-x+3$ ]]>$y=x+5$ ]]>$y=2x+6$ ]]>$y=-2x+2$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{3}-3{x}^{2}-9x+2\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ for $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-2\le x\le 1\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$. ]]> $7$ ]]>$2$ ]]>$-9$ ]]>$0$ ]]>$\frac{15}{2}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\int }_{1}^{2}{x}^{2}+5x-1\phantom{\rule{0.3em}{0ex}}dx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$. ]]> $\frac{23}{6}$ ]]>$\frac{53}{6}$ ]]>$\frac{17}{3}$ ]]>$9$ ]]>$\frac{55}{6}$ ]]> $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\int sin2x\phantom{\rule{0.3em}{0ex}}dx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$. ]]> $cos2x+C$ ]]>$-2cos2x+C$ ]]>$\frac{cos2x}{2}+C$ ]]>$2cos2x+C$ ]]>$-\frac{cos2x}{2}+C$ ]]>