Directional Derivative Calculator

The calculator will find the directional derivative (with steps shown) of the given function at the point in the direction of the given vector.

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.

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