Directional Derivative Calculator

This Directional Derivative Calculator helps you find the directional derivative of a multivariable function at a given point. The calculator first finds the gradient vector, then converts the given direction into a unit vector, and finally takes their dot product to get the result. This shows the rate of change of the function in the chosen direction. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Step-by-step method

1. Identify the function, the point, and the direction vector.
2. Compute each partial derivative with substeps.
3. Combine them into the gradient vector ∇f.
4. Evaluate ∇f at the point.
5. Convert the direction vector into a unit vector.
6. Use Dûf = ∇f(a) · û.

Formulas:

Example 1: f(x,y)=yx^2+sin(xy); (1,2); <3,4>

Step 1A - Identify the function.

In this problem: Read the given function.

Step 1B - Identify the point.

In this problem: Read the point where the directional derivative is required.

Step 1C - Identify the direction vector.

In this problem: Read the given direction vector.

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2B - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2C - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2D - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2E - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2F - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2G - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3B - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3C - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3D - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3E - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3F - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3G - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 4 - Combine into the gradient vector.

In this problem: Put the partial derivatives into ⟨ … ⟩.

Step 5A - Substitute the point into fx.

In this problem: Substitute the point values.

Step 5B - Simplify fx at the point.

In this problem: Simplify the substituted expression.

Step 5C - Substitute the point into fy.

In this problem: Substitute the point values.

Step 5D - Simplify fy at the point.

In this problem: Simplify the substituted expression.

Step 6 - Gradient at the point.

In this problem: Put the evaluated components into ⟨ … ⟩.

Step 7A - Find the magnitude of the direction vector.

In this problem: Use the vector magnitude formula.

Step 7B - Simplify the magnitude.

In this problem: Simplify the square root.

Step 8 - Convert the direction vector into a unit vector.

In this problem: Divide the direction vector by its magnitude.

Step 9A - Use the directional derivative formula.

In this problem: Substitute the gradient at the point and the unit vector.

Step 9B - Expand the dot product.

In this problem: Multiply matching components and add.

Step 9C - Simplify the result.

In this problem: Write the final directional derivative.

Final answer: Directional derivative = 2*cos(2) + 16/5 ≈ 2.37

Example 2: f(x,y)=x^2+y^2; (1,1); <1,1>

Step 1A - Identify the function.

In this problem: Read the given function.

Step 1B - Identify the point.

In this problem: Read the point where the directional derivative is required.

Step 1C - Identify the direction vector.

In this problem: Read the given direction vector.

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

Step 2B - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

Step 2C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

Step 3C - Combine the results.

In this problem: Add the derivatives from the previous substeps.

Step 4 - Combine into the gradient vector.

In this problem: Put the partial derivatives into ⟨ … ⟩.

Step 5A - Substitute the point into fx.

In this problem: Substitute the point values.

Step 5B - Simplify fx at the point.

In this problem: Simplify the substituted expression.

Step 5C - Substitute the point into fy.

In this problem: Substitute the point values.

Step 5D - Simplify fy at the point.

In this problem: Simplify the substituted expression.

Step 6 - Gradient at the point.

In this problem: Put the evaluated components into ⟨ … ⟩.

Step 7A - Find the magnitude of the direction vector.

In this problem: Use the vector magnitude formula.

Step 7B - Simplify the magnitude.

In this problem: Simplify the square root.

Step 8 - Convert the direction vector into a unit vector.

In this problem: Divide the direction vector by its magnitude.

Step 9A - Use the directional derivative formula.

In this problem: Substitute the gradient at the point and the unit vector.

Step 9B - Expand the dot product.

In this problem: Multiply matching components and add.

Step 9C - Simplify the result.

In this problem: Write the final directional derivative.

Final answer: Directional derivative = 2*sqrt(2) ≈ 2.83