(At one time, Maple's Smart Pop-Up applied to ${\tan }^2\left(x^2\right)$ sufficed to convert the derivative into the form obtained in Table 2.7.3(a). At the present time, further progress towards this goal requires the application of commands.)

$\frac{\mathit{\ⅆ}}{\mathit{\ⅆ}x} \left(\sin \left(x\right)\tan \left(x^2\right)\right)$

$\cos \left(x\right)\tan \left(x^2\right)+2\sin \left(x\right)x\left(1+{\tan \left(x^2\right)}^2\right)$

$\stackrel{\text{simplify}}{\=}$

$\frac{\cos \left(x\right)\sin \left(x^2\right)\cos \left(x^2\right)+2\sin \left(x\right)x}{{\cos \left(x^2\right)}^2}$

$\stackrel{\text{expand}}{\=}$

$\frac{\cos \left(x\right)\sin \left(x^2\right)}{\cos \left(x^2\right)}+\frac{2\sin \left(x\right)x}{{\cos \left(x^2\right)}^2}$

$\frac{d}{\mathrm{dx}} \left(\sin \left(x\right)\tan \left(x^2\right)\right)$

$\=\sin \left(x\right)\cdot \frac{d}{\mathrm{dx}}\tan \left(x^2\right)\+\tan \left(x^2\right)\cdot \frac{d}{\mathrm{dx}}\sin \left(x\right)$

$\=\sin \left(x\right)\cdot {\sec }^2\left(x^2\right)\cdot \frac{d}{\mathrm{dx}}\left(x^2\right)\+\tan \left(x^2\right)\cdot \cos \left(x\right)$

$\=2 x \sin \left(x\right) {\sec }^2\left(x^2\right)\+\cos \left(x\right) \tan \left(x^2\right)$

Table 2.7.3(a)   Stepwise evaluation of the derivative of $\left(\sin \left(x\right)\tan \left(x^2\right)\right)$