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  2. Rekenmachines
  3. Rekenmachines: Analyse III
  4. Rekenmachine voor differentiaal- en integraalrekening

Rekenmachine voor de richtingsafgeleide

Bereken richtingsafgeleiden stap voor stap

De rekenmachine zal de richtingsafgeleide (met uitgewerkte stappen) van de gegeven functie in het punt, in de richting van de gegeven vector bepalen.

Enter a function:
Enter a point:
Enter a point, for example, `(1, 2, 3)` as `x,y,z=1,2,3`, or simply `1,2,3`, if you want the order of variables to be detected automatically.
Enter vector $$$\vec{u}$$$: ()
As comma-separated coordinates, for example, `2i-3j` should be entered as 2,-3.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: find the directional derivative of $$$e^{x} + \sin{\left(y z \right)}$$$ at $$$\left(x,y,z\right)=\left(3,0,\frac{\pi}{3}\right)$$$ in the direction of the vector $$$\vec{u}=\left(2,3,6\right)$$$

Find the gradient of the function and evaluate it at the given point:

$$$\nabla \left(e^{x} + \sin{\left(y z \right)}\right)|_{\left(x,y,z\right)=\left(3,0,\frac{\pi}{3}\right)} = \left(e^{3},\frac{\pi}{3},0\right)$$$ (for steps, see gradient calculator)

Find the length of the vector: $$$\left|\vec{u}\right|=\sqrt{\left(2\right)^2+\left(3\right)^2+\left(6\right)^2}=7$$$

To normalize the vector, divide each component by the length:

$$$\vec{u}$$$ becomes $$$\left(\frac{2}{7},\frac{3}{7},\frac{6}{7}\right)$$$.

Finally, the directional derivative is the dot product of the gradient and the normalized vector:

$$$D\left(e^{x} + \sin{\left(y z \right)}\right)_{\vec{u}}\left(3,0,\frac{\pi}{3}\right)=\left(e^{3},\frac{\pi}{3},0\right) \cdot \left(\frac{2}{7},\frac{3}{7},\frac{6}{7}\right) = \frac{\pi + 2 e^{3}}{7}$$$ (for steps, see dot product calculator)

Answer: $$$D\left(e^{x} + \sin{\left(y z \right)}\right)_{\vec{u}}\left(3,0,\frac{\pi}{3}\right)=\frac{\pi + 2 e^{3}}{7} \approx 6.1875237857093$$$