# Directional Derivative Calculator

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## Directional Derivative Calculator with steps

Directional derivative calculator is used to find the gradient and directional derivative of the given function. It takes the points of x & y coordinates along with the points of the vector. It is a type of derivative calculator.

## How to use this directional derivative calculator?

Follow the below steps to find the directional derivatives of the functions.

• Input the multivariable function.
• To enter the math keys, hit the keypad icon.
• Write the values of $$U_1\&U_2$$.
• Enter the x & y coordinates.
• Click the calculate button.
• To recalculate hit the clear button.

## What is the directional derivative?

In calculus, the directional derivative of a multivariable differentiable function along with a vector v at the given point x intuitively represents the instantaneous rate of change of function, moving through x with velocity specified by v.

The directional derivative of a scalar function f(x) along with the vector v is a function $$∇_v\:f$$ defined by the limit

$$∇_v\:f\left(x\right)=\lim _{h\to 0}\left(\frac{f\left(x+hv\right)-f\left(x\right)}{h}\right)$$

The directional derivative used various notations such as:

$$∇_v\:f\left(x\right),\:f_v'\left(x\right),\:\partial \:_vf\left(x\right),\:v.∇f\left(x\right),\:or\:v.\frac{\partial \:f\left(x\right)}{\partial \:x}$$

### The formula of the directional derivative.

The directional derivative is the dot product of the gradient and the normalized vector.

$$∇_v\left(f\left(x\right)\right)=∇f\left(x\right).\:\frac{v}{\left|v\right|}$$

## Rules of the directional derivative

Below are some rules of directional derivatives.

 Rule Name Rules Sum rule $$∇_v\left(f\left(x\right)+g\left(x\right)\right)\:=∇_v\:f\left(x\right)+∇_v\:g\left(x\right)$$ Difference rule $$∇_v\left(f\left(x\right)-g\left(x\right)\right)\:=∇_v\:f\left(x\right)-∇_v\:g\left(x\right)$$ Constant factor rule $$∇_v\left(cf\left(x\right)\right)\:=c∇_v\:f\left(x\right)$$ Product rule $$∇_v\left(f\left(x\right)\cdot g\left(x\right)\right)\:=g\left(x\right)∇_v\:f\left(x\right)+f\left(x\right)∇_v\:g\left(x\right)$$

## How to calculate the directional derivative?

Following is a solved example of a directional derivative.

Example

Find the directional derivative of $$e^x+3y$$ at (x, y) = (3, 4) along with the vector u = (1, 2).

Solution

Step 1: Write the given function with the gradient notation.

$$∇\left(f\left(x,y\right)\right)=\frac{\partial f\left(x,y\right)}{\partial x},\frac{\partial f\left(x,y\right)}{\partial y}$$

$$∇\left(e^x+3y\right)=\frac{\partial }{\partial x}\left(e^x+3y\right),\frac{\partial \:}{\partial y}\left(e^x+3y\right)$$

Step 2: Take the partial derivative of the above function with respect to x & y.

$$\frac{\partial \:}{\partial x}\left(e^x+3y\right)=e^x$$

$$\frac{\partial \:}{\partial y}\left(e^x+3y\right)=3$$

Step 3: Put the given points of x & y.

$$∇\left(e^x+3y\right)|_{\left(x,y\right)=\left(3,4\right)}=\left(e^3,3\right)$$

Step 4: Find the length of the given vector and normalize the vector.

$$|u⃗\:|=\sqrt{1^2+2^2}=\sqrt{5}$$

To normalize divide (1, 2) component by $$\sqrt{5}$$.

$$\left(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)$$

Step 5: Take the dot product of the gradient and the normalized vector.

$$D_{u⃗\:}\left(e^x+3y\right)|_{\left(3,4\right)}\:=\left(e^3,3\right).\left(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)$$

$$D_{u⃗\:}\left(e^x+3y\right)|_{\left(3,4\right)}\:=\frac{e^3+6}{\sqrt{5}}$$

### References

Wikimedia Foundation. (2022, March 5) | what is the directional derivative? | Wikipedia.

Example of directional derivative | Calculus III - directional derivatives (practice problems).