Gradient Calculator

Gradient Calculator with steps 

Gradient calculator is used to calculate the gradient of two or three points of a vector line by taking the partial derivative of the function. This calculator provides the solution with steps.

What is the gradient?

In calculus, the gradient is an operator of the differential that is applied to a vector-valued function to produce a vector whose components are the partial derivatives of the multivariable function w.r.t its variables. 

The gradient is just like a slope. It is denoted by the “∇” symbol. It is applied to the multivariable functions. 

Gradient formula

The formula of the gradient is:

∇ f(x, y, z) = [∂f/∂x ∂f/∂y ∂f/∂z]

How to calculate gradient?

Here are a few solved examples of the gradient to learn how to calculate it.

Example 1: For two points

Find the gradient of 2x² – 3y³ for points (4, 5).

Solution 

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

∇ f(x, y) = ∇ (2x² – 3y³)

Step 2: Now take the formula of the gradient and solve the above function.

∇ f(x, y) = [∂f/∂x, ∂f/∂y]

∇ (2x² – 3y³) = [∂(2x² – 3y³)/∂x, ∂(2x² – 3y³) /∂y]

∇ (2x² – 3y³) = [(4x^2-1 – 0), (0 – 9y^3-1)]

∇ (2x² – 3y³) = [4x, – 9y²]

Step 3: Now substitute the given points.

∇ (2x² – 3y³) |_{(x, y)} = [4x, – 9y²]

∇ (2x² – 3y³) |_{(x, y) = (4, 5)} = [4(4), – 9(5)²]

∇ (2x² – 3y³) |_{(x, y) = (4, 5)} = [16, – 9(25)]

∇ (2x² – 3y³) |_{(x, y) = (4, 5)} = [16, – 225]

Example 2: For three points

Find the gradient of 3x³ + 4y² + 3z³ for points (2, 3, 4).

Solution 

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

∇ f(x, y, z) = ∇ (3x³ + 4y² + 3z³)

Step 2: Now take the formula of the gradient and solve the above function.

∇ f(x, y, z) = [∂f/∂x, ∂f/∂y, ∂f/∂z]

∇ (3x³ + 4y² + 3z³) = [∂(3x³ + 4y² + 3z³)/∂x, ∂(3x³ + 4y² + 3z³) /∂y, ∂(3x³ + 4y² + 3z³) /∂z]

∇ (3x³ + 4y² + 3z³) = [(9x^3-1 + 0 + 0), (0 + 8y^2-1 + 0), (0 + 0 + 9z^3-1)]

∇ (3x³ + 4y² + 3z³) = [9x², 8y, 9z²]

Step 3: Now substitute the given points.

∇ (3x³ + 4y² + 3z³) |_{(x, y, z)} = [9x², 8y, 9z²]

∇ (3x³ + 4y² + 3z³) |_{(x, y, z) = (2, 3, 4)} = [9(2)², 8(3), 9(4)²]

∇ (3x³ + 4y² + 3z³) |_{(x, y, z) = (2, 3, 4)} = [9(4), 8(3), 9(16)]

∇ (3x³ + 4y² + 3z³) |_{(x, y, z) = (2, 3, 4)} = [36, 24, 144]