Directional Derivative Calculator

An online directional derivative calculator determines the directional derivative and gradient of a function at a given point of a vector. Also, this free calculator shows you the step-by-step calculations for the particular points.

Now, let's see how to find directional derivatives using formulas and examples.

What is Directional Derivative?

In mathematics, it is intuitive to derive in the direction of the multidimensional differential function of a given vector v at a given point x. It is the instantaneous rate of change of a function moving in x with a velocity determined by v. Therefore, the generalized concept of partial derivatives, in which the rate of change is obtained along with one of the curvilinear coordinate curves. While all other coordinates remain constant.

(Image)

Directional Derivative Formula:

Let f be a curve whose tangent vector at some chosen point is v. The directional derivative calculator find a function f for p may be denoted by any of the following:

• \(\ ∇_pf(x)\)
• \(\ f’_p(x)\)
• \(\ D_pf(x)\)
• \(\ Df(x)(p)\)
• \(\ ∇f(x)\)

So, directional derivative of the scalar function is:

\(\ f(x) =\ f(x_1, x_2, …., x_{n-1}, x_n)\) with the vector \(\ v = (v_1, v_2, …, v_n)\) is the function \(\ ∇_vf\), which is calculated by

\(\ ∇_vf(x) = lim f(x + hv) – f(x)/ h\)

This is the formula used by the directional derivative calculator to find the derivative of a given function.

The Gradient:

The gradient ∇f is the vector pointing to the direction of the greatest upward slope, and its length is the directional derivative in this direction, and the directional derivative is the dot product between the gradient and the unit vector:

\(\ M_uf = ∇f⋅u\)

Directional Derivative Example:

If \(\ z=14−x^2−y^2\) and let M=(3,4). Find the directional derivative of f, at M, in the following directions:

• Toward the point N=(5,6),
• in the direction of ⟨2,−1⟩, and
• toward the origin.

Answer:

The point M=(3,4)is indicated in the x,y-plane as well as the point (3,4,9)which lies on the surface of "f". We find by using directional derivative formula

 fx(x,y)=−2x and fx(3,4)=−2,

 f_y(x,y)=−2yand f_y(1,2)=−4

Let \(\vec u_1\) be the unit vector that points from the point (3,4) to the point Q=(3,4). The vector \(\vec PQ=(2,2)\) the vector in this direction is \(\vec u_1=(\frac{1}{\sqrt{2}})\).

Thus the directional derivative of f at (3,4) in the direction of

\(\vec u_1\ is\ \vec Du_1\ f(1,2)=−2(\frac{1}{sqrt{2}})+(−4)(\frac{1}{\sqrt{2}})=\frac{−6}{\sqrt{2}}≈−4.24\).

The rate of change of an object is moving from the point (3,4,9) on the surface in the direction of \(\mathbf{u}_1\) (which points toward the point Q) is about −4.24.

How Directional Derivative Calculator Works?

Use this online calculator to find the gradient points and directional derivative of a given function with these steps:

Input:

• First of all, select how many points are required for the direction of a vector.
• Now, to find the directional derivative, enter a function.
• Then, enter the given values for points and vectors.
• To continue the process, click the calculate button.

Output:

• The directional derivative calculator computes the derivatives of a given function in the direction of given vectors.
• It calculates the gradient by taking the derivative of a function concerning each variable.

FAQ:

What is a direction gradient?

If the gradient of the function at the point “p” is not zero, the direction of the gradient is the direction in which the function of p quickly increases, and the magnitude of the gradient is the growth rate in this direction.

Discuss the difference between gradient and directional derivative?

The directional derivative is the rate of change of a function in a given direction. The gradient can be used in the formula to determine the directional derivative. The gradient represents the direction of the maximum directional derivative in a function of more than one variable.

Is the first-order derivative the gradient?

The first-order derivative basically gives the direction. In other words, it tells us whether the function is increasing or decreasing. The first-order derivative can be interpreted as the instantaneous rate of change. This derivative can also be interpreted as the slope of the tangent.

Why is the derivation of the direction important?

In mathematics, it is intuitive to derive in the direction of the multidimensional differential function of a given vector v at a given point x. It is the instantaneous rate of change of a function, moving at x with the velocity determined by v. Directional derivation is a special case of Gateaux derivation.

In which direction is the directional derivative the largest?

When theta θ= 0, the directional derivative has the largest positive value. Therefore, the direction of maximum increase of function f coincides with the direction of the gradient vector. When θ = pi (or 180 degrees), the directional derivative takes the largest negative value.

Can directional derivatives be negative?

Yes, the directional derivative is the change in the direction, which can be positive, negative, or zero. The directional derivative is negative means that the function decreases in this direction or increases in the opposite direction.

Conclusion:

An online directional derivative calculator generalizes the partial derivatives to determine the slope in any direction and calculates the derivatives and gradients in three dimensions. You need a graph paper to find the directional derivative and vectors, but it also increases the chance of errors. So, use this free online calculator for finding the directional derivatives, which provides a step-wise solution with 100% accuracy.

Reference:

From the source of Wikipedia: Directional derivative, Notation, Definition, Using the only direction of the vector, Restriction to a unit vector.

From the source of Libre Text: Definition Directional Derivatives, theorem Directional Derivatives, The Gradient and Directional Derivatives.

From the source of Active Calculus: Directional Derivatives and the Gradient, The Direction of the Gradient, The Length of the Gradient, Natural Applications.

From the source of Libre Text: Directional Derivatives and the Gradient, Theorem for Directional Derivatives, Theorem for the gradient.