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Directional Derivative Calculator

Calculate directional derivatives step by step

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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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