Question

Find the first partial derivatives of the function.

z = (6x + 2y)^9

dz/dx=
dz/dy=

Answer 1

To find the first partial derivatives of the given function, we differentiate with respect to x and y separately. The derivative dz/dx is 54(6x + 2y)^8, and the derivative dz/dy is 18(6x + 2y)^8.

Step-by-step explanation:

To find the first partial derivatives of the function z = (6x + 2y)^9, we need to differentiate with respect to x and y separately. Let's start with dz/dx:

Using the power rule, we bring down the exponent and multiply it by the coefficient of x. The derivative of 6x + 2y with respect to x is 6, so we have 9(6x + 2y)^8 * 6 = 54(6x + 2y)^8.

Now, let's find dz/dy:

Since the derivative of 6x + 2y with respect to y is 2, we multiply 9(6x + 2y)^8 by 2, giving us 18(6x + 2y)^8.

Answer 2

If z = (6x + 2y)⁹, then the partial derivatives of z are

• with respect to x :

∂z/∂x = 9 (6x + 2y)⁹⁻¹ • ∂(6x + 2y)/∂x

∂z/∂x = 9 (6x + 2y)⁸ • 6

∂z/∂x = 54 (6x + 2y)⁸

• with respect to y :

∂z/∂y = 9 (6x + 2y)⁹⁻¹ • ∂(6x + 2y)/∂y

∂z/∂y = 9 (6x + 2y)⁸ • 2

∂z/∂y = 18 (6x + 2y)⁸