Problem:
Differentiate the following function:
$\sqrt{7 x+6}-\sqrt{3} \sqrt{x}$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Possible derivation:} \\
 \frac{d}{dx}\left(-\left(\sqrt{3} \sqrt{x}\right)+\sqrt{6+7 x}\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Differentiate }\text{the }\text{sum }\text{term }\text{by }\text{term }\text{and }\text{factor }\text{out }\text{constants:} \\
  \text{= }-\sqrt{3} \left(\frac{d}{dx}\left(\sqrt{x}\right)\right)+\frac{d}{dx}\left(\sqrt{6+7 x}\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Use }\text{the }\text{power }\text{rule, }\frac{d}{dx}\left(x^n\right)=n x^{n-1}, \text{where }n=\frac{1}{2}.\text{$\backslash $n}\frac{d}{dx}\left(\sqrt{x}\right)=\frac{d}{dx}\left(x^{1/2}\right)=\frac{x^{-1/2}}{2}: \\
  \text{= }\frac{d}{dx}\left(\sqrt{6+7 x}\right)-\sqrt{3} \fbox{$\frac{1}{2 \sqrt{x}}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Using }\text{the }\text{chain }\text{rule, }\frac{d}{dx}\left(\sqrt{7 x+6}\right)=\frac{d\sqrt{u}}{du} \frac{du}{dx}, \text{where }u=7 x+6 \text{and }\frac{d\text{}}{du}\left(\sqrt{u}\right)=\frac{1}{2 \sqrt{u}}: \\
  \text{= }-\frac{\sqrt{3}}{2 \sqrt{x}}+\fbox{$\frac{\frac{d}{dx}(6+7 x)}{2 \sqrt{6+7 x}}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Differentiate }\text{the }\text{sum }\text{term }\text{by }\text{term }\text{and }\text{factor }\text{out }\text{constants:} \\
  \text{= }-\frac{\sqrt{3}}{2 \sqrt{x}}+\fbox{$\frac{d}{dx}(6)+7 \left(\frac{d}{dx}(x)\right)$} \frac{1}{2 \sqrt{6+7 x}} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{derivative }\text{of }6 \text{is }\text{zero}: \\
  \text{= }-\frac{\sqrt{3}}{2 \sqrt{x}}+\frac{7 \left(\frac{d}{dx}(x)\right)+\fbox{$0$}}{2 \sqrt{6+7 x}} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Simplify }\text{the }\text{expression}: \\
  \text{= }-\frac{\sqrt{3}}{2 \sqrt{x}}+\frac{7 \left(\frac{d}{dx}(x)\right)}{2 \sqrt{6+7 x}} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{derivative }\text{of }x \text{is }1: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} &  \text{= }-\frac{\sqrt{3}}{2 \sqrt{x}}+\fbox{$1$} \frac{7}{2 \sqrt{6+7 x}} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/calculus/derivatives_w_steps/609.txt'}