 To find the gradient, enter the multivariable function, points of line, and click calculate button using gradient calculator

for( x, y )
for( x, y, z )

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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.

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.

The formula of the gradient is:

∇ f(x, y, z) = [∂f/∂x ∂f/∂y ∂f/∂z]

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) = [(4x2-1 – 0), (0 – 9y3-1)]

∇ (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) = [(9x3-1 + 0 + 0), (0 + 8y2-1 + 0), (0 + 0 + 9z3-1)]

∇ (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]

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