Answer
$f^{\prime}(x)=(8x+3)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}^{2}(4x^{2}+3x)$
Work Step by Step
Apply Th. 5.18:
Let $u$ be a differentiable function of $x$.
$\displaystyle \frac{d}{dx}[\tanh u]=(\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}^{2}u)u^{\prime}$
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$\displaystyle \frac{d}{dx}\tanh(4x^{2}+3x)=\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}^{2}(4x^{2}+3x)[4x^{2}+3x]^{\prime}$
$=\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}^{2}(4x^{2}+3x)\cdot (8x+3)$
$f^{\prime}(x)=(8x+3)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}^{2}(4x^{2}+3x)$
