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### Section 3.9 : Chain Rule

14. Differentiate $$S\left( w \right) = \sqrt {7w} + {{\bf{e}}^{ - w}}$$ .

Hint : Don’t get too locked into problems only requiring a single use of the Chain Rule. Sometimes separate terms will require different applications of the Chain Rule, or maybe only one of the terms will require the Chain Rule.
Show Solution

For this problem each term will require a separate application of the Chain Rule and make sure you are careful with parenthesis in dealing with the root in the first term.

The derivative is then,

$S\left( w \right) = {\left( {7w} \right)^{\frac{1}{2}}} + {{\bf{e}}^{ - w}}\hspace{0.25in}\,\, \Rightarrow \hspace{0.25in}S'\left( w \right) = \frac{1}{2}\left( 7 \right){\left( {7w} \right)^{ - \,\,\,\frac{1}{2}}} - {{\bf{e}}^{ - w}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{7}{2}{{\left( {7w} \right)}^{ - \,\,\,\frac{1}{2}}} - {{\bf{e}}^{ - w}}}}$