(1) Partial derivatives of a polynomial

Determine the partial derivatives of the function
\( f(x,y)= 3x^2y + 9xy^2 + 1x + 6y + 5 \).
Calculate \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

(2) Partial derivatives at a point

A function is given
\( f(x,y)= 3x^2 + -2xy + 1y^2 + -5x + -3y \).
Calculate the first-order partial derivatives and their values at the point \( (x,y)=(-2, 1) \).

(3) Partial derivatives of a third-degree function

Determine the partial derivatives of the function
\( f(x,y)= 5x^3 + 4x^2y + 7xy^2 + 1y^3 + 6xy \).
Calculate \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

(4) Partial derivative of a rational function

A function is given
\( f(x,y)= \frac{2x^2 + 8y}{6x + 5y} \).
Determine the partial derivative \( \frac{\partial f}{\partial x} \).

(5) Partial derivatives of a power function

Determine the partial derivatives of the function
\( f(x,y)= 5x^{1}y^{2} + 6xy + 4y \).
Calculate \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

(6) Partial derivatives of an exponential function

A function is given
\( f(x,y)= e^{5x + 1y} + 2xy + 3x \).
Determine the first-order partial derivatives.

(7) Partial derivatives of a trigonometric function

Determine the partial derivatives of the function
\( f(x,y)= 5\sin(xy) + 1x\cos(y) + 3y + 4 \).
Calculate \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).

(8) Partial derivatives of a logarithmic function

A function is given
\( f(x,y)= \ln(3x + 5y) + 6xy + 7x \).
Determine the first-order partial derivatives.

(9) Second-order partial derivatives

A function is given
\( f(x,y)= 6x^2y + 4xy^2 + 1x + 2y \).
Calculate the second partial derivatives \( \frac{\partial^2 f}{\partial x^2} \), \( \frac{\partial^2 f}{\partial y^2} \) and \( \frac{\partial^2 f}{\partial x \partial y} \).

(10) Partial derivative of a composite function

A function is given
\( f(x,y)= ( 2x + 3y )^{5} + 1xy \).
Determine the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).