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