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### Section 3.9 : Chain Rule

7. Differentiate $$f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}$$ .

Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
Show Solution

Note that we only need to use the Chain Rule on the second term as we can differentiate the first term without the Chain Rule.

Now, recall that for exponential functions outside function is the exponential function itself and the inside function is the exponent. The derivative is then,

$\require{bbox} \bbox[2pt,border:1px solid black]{{f'\left( t \right) = \left( {4 + 7{t^6}} \right){{\bf{e}}^{4t + {t^{\,7}}}}}}$