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Section 3.9 : Chain Rule
1. Differentiate \(f\left( x \right) = {\left( {6{x^2} + 7x} \right)^4}\) .
Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. The derivative is then,
\[f'\left( x \right) = 4{\left( {6{x^2} + 7x} \right)^3}\left( {12x + 7} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{4\left( {12x + 7} \right){{\left( {6{x^2} + 7x} \right)}^3}}}\]