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.
  • Press the show more button to view the steps.
  • 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.


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.


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


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.


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


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


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




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

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