Solve for ∂∂𝑥{∂𝑧∂𝑥} from 𝑧=3𝑥2+4𝑥𝑦-5𝑦2

To solve for ∂∂𝑥{∂𝑧∂𝑥} from 𝑧=3𝑥2+4𝑥𝑦-5𝑦2, we first need to find ∂𝑧/∂𝑥, then differentiate that result with respect to x.

Step 1: Find ∂𝑧/∂𝑥 The partial derivative of z with respect to x is found by differentiating z with respect to x, treating y as a constant.

∂𝑧/∂𝑥 = ∂/∂𝑥(3𝑥2) + ∂/∂𝑥(4𝑥𝑦) - ∂/∂𝑥(5𝑦2) = 6𝑥 + 4𝑦 - 0 = 6𝑥 + 4𝑦

Step 2: Find ∂∂𝑥{∂𝑧∂𝑥} Now we differentiate ∂𝑧/∂𝑥 with respect to x, treating y as a constant.

∂∂𝑥{∂𝑧∂𝑥} = ∂/∂𝑥(6𝑥 + 4𝑦) = 6 + 0 = 6

So, ∂∂𝑥{∂𝑧∂𝑥} = 6.