Question

Find the partial derivatives of the function f(x,y) = -4x^7y/(-7x-6y).

a) fx = 28x^6y/(7x + 6y)^2, fy = -4x^7/(7x + 6y)^2
b) fx = -4x^7/(7x + 6y)^2, fy = 28x^6y/(7x + 6y)^2
c) fx = 28x^6y/(7x - 6y)^2, fy = -4x^7/(7x - 6y)^2
d) fx = -4x^7/(7x - 6y)^2, fy = 28x^6y/(7x - 6y)^2

Answer 1

The correct partial derivatives for the function given are fx = 28x^6y/(7x + 6y)^2 and fy = -4x^7/(7x + 6y)^2, which corresponds to answer choice a).

Step-by-step explanation:

The student has asked to find the partial derivatives of the function f(x,y) = -4x7y/(-7x-6y). To find the partial derivative with respect to x, denoted as fx, we treat y as a constant and differentiate with respect to x. Similarly, to find the partial derivative with respect to y, denoted as fy, we treat x as a constant and differentiate with respect to y.

To calculate the partial derivative with respect to x (fx), we apply the quotient rule, getting fx = 28x6y/(7x + 6y)2. To calculate the partial derivative with respect to y (fy), we again use the quotient rule, resulting in fy = -4x7/(7x + 6y)2. Therefore, the correct answer is option a).

Answer 2

The partial derivation of this function specifies the b option as the best . Thus the correct option is b)
`fx = -4x^7/(7x + 6y)^2, fy = 28x^6y/(7x + 6y)^2`

Step-by-step explanation:

To find the partial derivatives of the given function
`\(f(x, y) = (-4x^7y)/(-7x-6y),\)`we use the quotient rule. The quotient rule states that for a function
`\(g(x) = (u(x))/(v(x)),\)` the derivative is given by
`\(g'(x) = (u'v - uv')/(v^2).\)`

First, let's find the partial derivative with respect to
`\(x\)`, denoted as
`\(f_x:\)`

`\[f_x = ((-4x^7y)'(-7x-6y) - (-4x^7y)(-7x-6y)')/((-7x-6y)^2).\]`

Simplifying, we get
`\(f_x = (-28x^6y)/((7x+6y)^2).\)`

Next, let's find the partial derivative with respect to
`\(y\),` denoted as
`\(f_y:\)`

`\[f_y = ((-4x^7y)'+(-7x-6y)(-4x^7))/((-7x-6y)^2).\]`

Simplifying, we get
`\(f_y = (28x^6y)/((7x+6y)^2).\)`

Therefore, the correct answer is b)
`\(fx = (-4x^7)/((7x + 6y)^2),\) \(fy = (28x^6y)/((7x + 6y)^2).\)`