To find the partial derivatives of the function f(x,y) = 3(9x - 4y + 5)^9, we can use the power rule for differentiation. The power rule states that for a function f(x) = (g(x))^n, the derivative f'(x) can be found by multiplying the exponent n by the derivative of g(x), which we denote as g'(x).
First, let's find f_x, the partial derivative of f(x,y) with respect to x.
To do this, we treat y as a constant and differentiate the function with respect to x. The terms that do not involve x will disappear since their derivative with respect to x is zero.
f_x = d/dx (3(9x - 4y + 5)^9)
= 9 * 3 * (9x - 4y + 5)^8 * d/dx (9x - 4y + 5)
= 27 * (9x - 4y + 5)^8 * d/dx (9x) - 27 * (9x - 4y + 5)^8 * d/dx (4y) + 27 * (9x - 4y + 5)^8 * d/dx (5)
= 27 * (9x - 4y + 5)^8 * 9 - 0 + 0
= 243 * (9x - 4y + 5)^8
Therefore, f_x(x,y) = 243 * (9x - 4y + 5)^8.
Next, let's find f_y, the partial derivative of f(x,y) with respect to y.
To do this, we treat x as a constant and differentiate the function with respect to y. The terms that do not involve y will disappear since their derivative with respect to y is zero.
f_y = d/dy (3(9x - 4y + 5)^9)
= 9 * 3 * (9x - 4y + 5)^8 * d/dy (9x - 4y + 5)
= 0 - 27 * (9x - 4y + 5)^8 * d/dy (4y) + 0
= -108 * (9x - 4y + 5)^8
Therefore, f_y(x,y) = -108 * (9x - 4y + 5)^8.
In summary:
- f_x(x,y) = 243 * (9x - 4y + 5)^8
- f_y(x,y) = -108 * (9x - 4y + 5)^8