${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)$

$\=\left[\begin{array}{c}2{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\\ 3{\ⅇ}^{\sin \left(6\right)}\left(\cosh \left(2\right)\cos \left(6\right)\+\sinh \left(2\right)\right)\\ 2{\ⅇ}^{\sin \left(6\right)}\left(\cosh \left(2\right)\cos \left(6\right)-\sinh \left(2\right)\right)\end{array}\right]·\left(\frac{1}{\sqrt{89}}\left[\begin{array}{r}2\\ -9\\ 2\end{array}\right]\right)$

$\=-\frac{1}{\sqrt{89}}{\ⅇ}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\+27\sinh \left(2\right)\right)$

Initialize

Loading Student:-MultivariateCalculus

$f\={\ⅇ}^{\sin \left(y z\right)}\cosh \left(\frac{x y}{z-y}\right)$$\stackrel{\text{assign}}{\to }$

$\mathbf{v}\=⟨2\,-9\,2⟩$$\stackrel{\text{assign}}{\to }$$$

$\mathbf{u}\=\mathbf{v}\/{∥\mathbf{v}∥}_2$$\stackrel{\text{assign}}{\to }$$$

Instant answer via the Context Panel

$f$

${\ⅇ}^{\sin \left(yz\right)}\cosh \left(\frac{xy}{z-y}\right)$

$\stackrel{\text{directional derivative}}{\to }$

$\frac{4}{89}{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\sqrt{89}-\frac{9}{89}\left(3\cos \left(6\right){\ⅇ}^{\sin \left(6\right)}\cosh \left(2\right)\+3{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\right)\sqrt{89}\+\frac{2}{89}\left(2\cos \left(6\right){\ⅇ}^{\sin \left(6\right)}\cosh \left(2\right)-2{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\right)\sqrt{89}$

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

$-\frac{1}{89}\sqrt{89}{\ⅇ}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\+27\sinh \left(2\right)\right)$

Obtain $\genfrac{}{}{0ex}{}{\left(\nabla f\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\mathrm{P}}$ 

${\ⅇ}^{\sin \left(y z\right)}\cosh \left(\frac{x y}{z-y}\right)$$\stackrel{\text{gradient}}{\to }$ $\stackrel{\text{select entry 1}}{\to }$ $\stackrel{\text{assign to a name}}{\to }$$\mathrm{Gf}$

Obtain ${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)\=\(\genfrac{}{}{0ex}{}{\left(\nabla f\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\mathrm{P}}\)·\mathbf{u}$ 

$\mathbf{Gf}·\mathbf{u}$

$\frac{4}{89}{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\sqrt{89}-\frac{9}{89}\left(3\cos \left(6\right){\ⅇ}^{\sin \left(6\right)}\cosh \left(2\right)\+3{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\right)\sqrt{89}\+\frac{2}{89}\left(2\cos \left(6\right){\ⅇ}^{\sin \left(6\right)}\cosh \left(2\right)-2{\ⅇ}^{\sin \left(6\right)}\sinh \left(2\right)\right)\sqrt{89}$

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

$-\frac{1}{89}\sqrt{89}{\ⅇ}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\+27\sinh \left(2\right)\right)$

Obtain $\mathbf{R}\=\mathbf{P}\+t \mathbf{u}$, the line through P with direction u

$\left[1\&comma;2\&comma;3\right]\&comma;\mathbf{u}$$\stackrel{\text{make line}}{\to }$$\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{representation}}{\to }$$\left[x\&equals;1\&plus;\frac{2}{89}t\sqrt{89}\&comma;y\&equals;2-\frac{9}{89}t\sqrt{89}\&comma;z\&equals;3\&plus;\frac{2}{89}t\sqrt{89}\right]$$\stackrel{\text{assign to a name}}{\to }$$L$

Obtain $w\left(t\right)\&equals;f\left(x\left(t\right)\&comma;y\left(t\right)\&comma;z\left(t\right)\right)$ and ${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)\&equals;w\prime \left(0\right)$

$\genfrac{}{}{0ex}{}{f}{\phantom{x\&equals;a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{L}$

${\&ExponentialE;}^{\sin \left(\left(2-\frac{9}{89}t\sqrt{89}\right)\left(3\&plus;\frac{2}{89}t\sqrt{89}\right)\right)}\cosh \left(\frac{\left(1\&plus;\frac{2}{89}t\sqrt{89}\right)\left(2-\frac{9}{89}t\sqrt{89}\right)}{1\&plus;\frac{11}{89}t\sqrt{89}}\right)$

$\stackrel{\text{differentiate w.r.t. t}}{\to }$

$\left(-\frac{9}{89}\sqrt{89}\left(3\&plus;\frac{2}{89}t\sqrt{89}\right)\&plus;\frac{2}{89}\left(2-\frac{9}{89}t\sqrt{89}\right)\sqrt{89}\right)\cos \left(\left(2-\frac{9}{89}t\sqrt{89}\right)\left(3\&plus;\frac{2}{89}t\sqrt{89}\right)\right){\&ExponentialE;}^{\sin \left(\left(2-\frac{9}{89}t\sqrt{89}\right)\left(3\&plus;\frac{2}{89}t\sqrt{89}\right)\right)}\cosh \left(\frac{\left(1\&plus;\frac{2}{89}t\sqrt{89}\right)\left(2-\frac{9}{89}t\sqrt{89}\right)}{1\&plus;\frac{11}{89}t\sqrt{89}}\right)\&plus;{\&ExponentialE;}^{\sin \left(\left(2-\frac{9}{89}t\sqrt{89}\right)\left(3\&plus;\frac{2}{89}t\sqrt{89}\right)\right)}\left(\frac{2}{89}\frac{\sqrt{89}\left(2-\frac{9}{89}t\sqrt{89}\right)}{1\&plus;\frac{11}{89}t\sqrt{89}}-\frac{9}{89}\frac{\left(1\&plus;\frac{2}{89}t\sqrt{89}\right)\sqrt{89}}{1\&plus;\frac{11}{89}t\sqrt{89}}-\frac{11}{89}\frac{\left(1\&plus;\frac{2}{89}t\sqrt{89}\right)\left(2-\frac{9}{89}t\sqrt{89}\right)\sqrt{89}}{{\left(1\&plus;\frac{11}{89}t\sqrt{89}\right)}^2}\right)\sinh \left(\frac{\left(1\&plus;\frac{2}{89}t\sqrt{89}\right)\left(2-\frac{9}{89}t\sqrt{89}\right)}{1\&plus;\frac{11}{89}t\sqrt{89}}\right)$

$\stackrel{\text{evaluate at point}}{\to }$

$-\frac{23}{89}\sqrt{89}{\&ExponentialE;}^{\sin \left(6\right)}\cos \left(6\right)\cosh \left(2\right)-\frac{27}{89}{\&ExponentialE;}^{\sin \left(6\right)}\sinh \left(2\right)\sqrt{89}$

$\stackrel{\text{simplify}}{\&equals;}$

$-\frac{1}{89}\sqrt{89}{\&ExponentialE;}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\&plus;27\sinh \left(2\right)\right)$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\&colon;$

$f≔{\&ExponentialE;}^{\sin \left(y z\right)}\cosh \left(\frac{x y}{z-y}\right)\&colon;$$$

$\mathbf{v}≔\left[2\&comma;-9\&comma;2\right]\&colon;$$$

Obtain the directional derivative

$\mathrm{simplify}\left(\mathrm{DirectionalDerivative}\left(f\&comma;\left[x\&comma;y\&comma;z\right]\&equals;\left[1\&comma;2\&comma;3\right]\&comma;\mathbf{v}\right)\right)$

$-\frac{1}{89}\sqrt{89}{\&ExponentialE;}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\&plus;27\sinh \left(2\right)\right)$

Compute ${\mathrm{D}}_{\mathbf{u}}f\left(\mathrm{P}\right)\&equals;\&lpar;\genfrac{}{}{0ex}{}{\left(\nabla f\right)}{\phantom{x\&equals;a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\mathrm{P}}\&rpar;·\mathbf{u}$ 

$\mathbf{Gf}\mathbf{≔}\mathrm{Gradient}\left(f\&comma;\left[x\&comma;y\&comma;z\right]\&equals;\left[1\&comma;2\&comma;3\right]\right)\left[\right]$

$$\begin{gathered} \mathbf{V}≔\mathrm{convert}\left(\mathbf{v}\&comma;\mathrm{Vector}\right)\&colon; \\ \mathbf{u}≔\mathrm{Normalize}\left(\mathbf{V}\right) \end{gathered}$$

$\mathrm{simplify}\left(\mathrm{DotProduct}\left(\mathbf{Gf}\&comma;\mathbf{u}\right)\right)$ = $-\frac{1}{89}\sqrt{89}{\&ExponentialE;}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\&plus;27\sinh \left(2\right)\right)$$$

Obtain the directional derivative from first principles

$L≔\mathrm{GetRepresentation}\left(\mathrm{Line}\left(\left[1\&comma;2\&comma;3\right]\&comma;\mathbf{u}\right)\&comma;\mathrm{form}\&equals;\mathrm{parametric}\right)$

$\left[x\&equals;1\&plus;\frac{2}{89}t\sqrt{89}\&comma;y\&equals;2-\frac{9}{89}t\sqrt{89}\&comma;z\&equals;3\&plus;\frac{2}{89}t\sqrt{89}\right]$

$w≔\mathrm{simplify}\left(\mathrm{eval}\left(f\&comma;L\right)\right)$

${\&ExponentialE;}^{-\sin \left(\frac{1}{7921}\left(9t\sqrt{89}-178\right)\left(2t\sqrt{89}\&plus;267\right)\right)}\cosh \left(\frac{1}{89}\frac{\left(2t\sqrt{89}\&plus;89\right)\left(9t\sqrt{89}-178\right)}{11t\sqrt{89}\&plus;89}\right)$

$\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{diff}\left(w\&comma;t\right)\&comma;t\&equals;0\right)\right)$ = $-\frac{1}{89}\sqrt{89}{\&ExponentialE;}^{\sin \left(6\right)}\left(23\cos \left(6\right)\cosh \left(2\right)\&plus;27\sinh \left(2\right)\right)$$$