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# Gradient Calculator

To find the gradient, enter the multivariable function, points of line, and click **calculate **button using 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 2x^{2 }– 3y^{3} for points (4, 5).

**Solution **

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

∇ f(x, y) = ∇ (2x^{2} – 3y^{3})

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

∇ f(x, y) = [∂f/∂x, ∂f/∂y]

∇ (2x^{2} – 3y^{3}) = [∂(2x^{2} – 3y^{3})/∂x, ∂(2x^{2} – 3y^{3}) /∂y]

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

∇ (2x^{2} – 3y^{3}) = [4x, – 9y^{2}]

**Step 3: **Now substitute the given points.

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

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

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

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

**Example 2: For three points**

Find the gradient of 3x^{3 }+ 4y^{2} + 3z^{3} for points (2, 3, 4).

**Solution **

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

∇ f(x, y, z) = ∇ (3x^{3} + 4y^{2} + 3z^{3})

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

∇ f(x, y, z) = [∂f/∂x, ∂f/∂y, ∂f/∂z]

∇ (3x^{3} + 4y^{2 }+ 3z^{3}) = [∂(3x^{3} + 4y^{2 }+ 3z^{3})/∂x, ∂(3x^{3} + 4y^{2 }+ 3z^{3}) /∂y, ∂(3x^{3} + 4y^{2 }+ 3z^{3}) /∂z]

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

∇ (3x^{3} + 4y^{2 }+ 3z^{3}) = [9x^{2}, 8y, 9z^{2}]

**Step 3: **Now substitute the given points.

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

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

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

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