Directional Derivative Calculator

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Directional Derivative Calculator

The directional derivative and gradient of a function at a particular point of a vector can be calculated using an online multivariable derivative calculator. This free gradient vector calculator also shows you how to calculate specific points step by step. Let's look at formulas and examples to discover how to find directional derivatives.

The direction of the multidimensional differential equation of a given vector v at a particular position x is intuitively deduced in mathematics. It's the instantaneous rate of change of a implicit function travelling in x with v as its velocity. All other coordinates, on the other hand, remain constant.

But don't be confused between directional deriative and implicit derivative as these both are performed on implicit function. The rules of implicit differentiation is other than driectional derivative. You can calculate implicit differentiation precisely using dy/dx calculator with steps.

Related: Find multivariable derivatives using direction vector calculator on this website.

Notations used by Directional Derivative Calculator

Let f be a curve with a tangent vector of v at a given location. Any of the following can be used in the directional derivative calculator to find a function f for p:

$$ ∇_p \; f(x) $$ $$ f_p'(x) $$ $$ D_p \; f(x) $$ $$ D \; f(x) \; (p) $$ $$ ∇ \; f(x) $$

The direction vector calculator uses these notations to compute the derivative of a function.

$$ ∇_v f(x) \;=\; lim \; f(x+hv) \;-\; \frac{f(x)}{h} $$

Related: Also find normal line equation calculator to solve for equation of line.

How Directional Derivative Calculator Works?

Follow these steps to get the gradient points and directional derivative of a given function using this online gradient vector calculator:


These are some simple steps for inputting values in the direction vector calculator in right way.

  • To calculate the directional derivative, Type a function for which derivative is required.
  • Now select f(x, y) or f(x, y, z).
  • Enter value for U1 and U2.
  • Type value for x and y co-ordinate.
  • Click the calculate button, to get output from multivariable derivative calculator.


The directional derivative calculator calculates a function's derivative in the direction of two vectors therefore, it is also known as vector derivative calculator. The gradient is calculated by taking the derivative for every variable's function inputted in the direction vector calculator.

Solved Example of Directional Derivative:

Find directional derivative of x2y + xy2 with respect to x and y, where U1= 2 and U2 = -3.


$$ \frac{36}{13} \; \approx \; 9.9846 $$


As the partial derivative calculator with steps is used to estimate the slope in a single given variable's direction only, but the derivatives and gradients are calculated in three dimensions using an online derivative calculator with steps which is called a directional derivative calculator.

Finding the directional derivative and vectors requires graph paper, but it also raises the risk of errors. But the vector derivative calculator makes it easy for us, now we get the directional derivatives, utilize this free online gradient vector calculator, which delivers a step-by-step solution with 100 percent accuracy.

If you are struggling to find the linear approximation of a given function, try differential approximation calculator. The online tool will instantly find the linear approximation step by step along with plot and possible intermediate steps.


What is the significance of the direction's derivation?

Answer: The direction of the multidimensional differential function of a specified vector v at a particular position x is intuitively deduced in mathematics. It's the instantaneous rate of change of a function travelling at x with v as its velocity. Gateaux derivation is a specific case of directional derivation.

Differentiate between a directional and a second derivative?

Answer: The rate of change of a function in a certain direction is called the directional derivative. The directional derivative can be calculated using the gradient in the formula. But the second derivative is the derivative of the derivative. It measures the instantaeous rate of change of first derivative of function. For such calculation use online higher order derivative calculator with steps for precise calculation.

What is the definition of a direction gradient?

Answer: The direction of the gradient is the direction in which the function of p quickly rises, where magnitude of the gradient is the growth rate, and if the gradient of the function at the point "p" is not zero.

Find directional derivative of x2y + xy2+ z2 with respect to x and y, where U1= 2, U2 = -3, and U3= -1 co-ordinate 2, -4, 3.


$$ 15 \sqrt{\frac{2}{7}} \approx 8.01874 $$

Is it possible for directional derivatives to be negative?

Answer: The directional derivative can be positive, negative, or zero as it is the change in direction. The function drops in this direction or grows in the opposite direction if the directional derivative is negative.

Alan Walker

Alan Walker

Last Updated August 26, 2022

I am Mathematician, Tech geek and a content writer. I love solving patterns of different math queries and write in a way that anyone can understand. Math and Technology has done its part and now its the time for us to get benefits from it.