Checking the math one last time!
Rules of Differentiation
Examples Coming Soon!
Basic Rules of Differentiation:
\[\frac{d}{dx}\left(c\right)=0\]
\[\frac{d}{dx}\left(x^{n}\right)=nx^{n-1}\]
\[\frac{d}{dx}\left(cf\left(x\right)\right)=c\frac{d}{dx}f\left(x\right)\]
\[\frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)\pm\frac{d}{dx}g\left(x\right)\]
\[\mbox{Product Rule: } \frac{d}{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)\]
\[\mbox{Quotient Rule: } \frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^{2}}\]
\(\mbox{Higher Order Derivatives:}\) \begin{array} {|r|r|}\hline \mbox{First Derivative} & y' \mbox{ or }\frac{dy}{dx} \\ \hline \mbox{Second Derivative} & y''\mbox{ or }\frac{d^{2}y}{dx^{2}} \\ \hline \mbox{Third Derivative} & y'''\mbox{ or }\frac{d^{3}y}{dx^{3}} \\ \hline \mbox{Fourth Derivative} & y^{(4)}\mbox{ or }\frac{d^{4}y}{dx^{4}} \\ \hline n^{th}\mbox{ Derivative} & y^{(n)}\mbox{ or }\frac{d^{n}y}{dx^{n}} \\ \hline \end{array}
\[\] Normal Line: the line perpendicular to the tangent line. \[m_{normal}=-\frac{1}{m_{tangent}}\]
Find \(\frac{dy}{dx}\) given that \(y=\left(3x^{2}+2x\right)\left(4x+1\right)\).
Find the equations of the tangent line and the normal line to the curve \(y=x^{2}\sqrt{x^{3}}\) at the point \((1,1)\).
Let \(u\) and \(v\) be function of \(x\) that are differentiable at \(x=-2\). Given that \(u(-2)=3\), \(u'(-2)=5\), \(v(-2)=-4\), and \(v'(-2)=2\), find the value of the following derivative at \(x=-2\).
\[\left.\frac{d}{dx}\right|_{x=-2}\left(6v+3uv\right)\]
Let \(u\) and \(v\) be function of \(x\) that are differentiable at \(x=4\). Given that \(u(4)=1\), \(u'(4)=-4\), \(v(4)=6\), and \(v'(4)=3\), find the value of the following derivative at \(x=4\).
\[\left.\frac{d}{dx}\right|_{x=4}\Bigg(\frac{v+2}{3u}+vx^{3}\Bigg)\]
Find the \(y'''\) of \(\sqrt{x^{3}}\).
Find \(f'\left(4\right)\) given that \(f\left(x\right)=\frac{x^{2}+3}{\sqrt{x}+2x}\).
For what values of \(x\) does the function \(f\left(x\right)=4x^{3}+15x^{2}-72x+6\) have a horizonatal tangent line?
Find the points on the graph of \(f\left(x\right)=x^{2}+3\) where the tangent line intersects the point \((0,-1)\).
Therefore, the tangent line at \((2,7)\) and \((-2,7)\) will intersect the point \((0,-1)\).
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Using your calculator, find the derivative of \(3x^2+16x+2\) at \(x=2\).
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Using your calculator, graph the function \(3x^2+16x+2\).
Now find the derivative using method #2.
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