Single Variable Calculus

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Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 48

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Differentiate $\displaystyle f(t) = \sin^2 (e^{\sin^2 t})$

$ \begin{equation} \begin{aligned} f'(t) =& \frac{d}{dt} [\sin ^2 (e^{\sin ^2 t})] \\ \\ f'(t) =& \frac{d}{dt} [\sin (e^{\sin ^2 t})]^2 \\ \\ f'(t) =& 2 \sin (e^{\sin ^2 t}) \frac{d}{dt} [\sin (e^{\sin ^2 t})] \\ \\ f'(t) =& 2 \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) \frac{d}{dt} (e^{\sin ^2 t}) \\ \\ f'(t) =& 2 \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) e^{\sin ^2 t} \frac{d}{dt} (\sin ^2 t) \\ \\ f'(t) =& 2 e^{\sin ^2 t} \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) (2 \sin t) \frac{d}{dt} (\sin t) \\ \\ f'(t) =& 4 e^{\sin ^2 t} \sin t \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) \cos t \end{aligned} \end{equation} $

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