Let's compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for each function.

(a) $f(x,y) = \tan(x^2 + y^2)$

Partial derivative with respect to $x$:

$\frac{\partial f}{\partial x} = \sec^2(x^2 + y^2) \cdot \frac{d}{dx}(x^2 + y^2)$ $= \sec^2(x^2 + y^2) \cdot 2x$

Partial derivative with respect to $y$:

$\frac{\partial f}{\partial y} = \sec^2(x^2 + y^2) \cdot \frac{d}{dy}(x^2 + y^2)$ $= \sec^2(x^2 + y^2) \cdot 2y$

(b) $f(x,y) = x e^{x^2 y}$

Partial derivative with respect to $x$:

Using the product rule: $\frac{\partial f}{\partial x} = e^{x^2 y} + x \cdot \frac{d}{dx}(e^{x^2 y})$ Using chain rule: $\frac{d}{dx} e^{x^2 y} = e^{x^2 y} \cdot (2xy)$ $\frac{\partial f}{\partial x} = e^{x^2 y} + x (e^{x^2 y} \cdot 2xy)$ $= e^{x^2 y} (1 + 2x^2 y)$

Partial derivative with respect to $y$:

$\frac{\partial f}{\partial y} = x \cdot \frac{d}{dy} e^{x^2 y}$ Using chain rule: $\frac{d}{dy} e^{x^2 y} = e^{x^2 y} \cdot (x^2)$ $\frac{\partial f}{\partial y} = x e^{x^2 y} x^2$ $= x^3 e^{x^2 y}$

(c) $f(x,y) = x^2 \ln(y^2)$

Partial derivative with respect to $x$:

$\frac{\partial f}{\partial x} = 2x \ln(y^2)$

Partial derivative with respect to $y$:

Using chain rule: $\frac{d}{dy} \ln(y^2) = \frac{1}{y^2} \cdot 2y = \frac{2}{y}$ $\frac{\partial f}{\partial y} = x^2 \cdot \frac{2}{y}$ $= \frac{2x^2}{y}$

Final Answers:

1. (a)

   • $\frac{\partial f}{\partial x} = 2x \sec^2(x^2 + y^2)$
   • $\frac{\partial f}{\partial y} = 2y \sec^2(x^2 + y^2)$

2. (b)

   • $\frac{\partial f}{\partial x} = e^{x^2 y} (1 + 2x^2 y)$
   • $\frac{\partial f}{\partial y} = x^3 e^{x^2 y}$

3. (c)

   • $\frac{\partial f}{\partial x} = 2x \ln(y^2)$
   • $\frac{\partial f}{\partial y} = \frac{2x^2}{y}$

Would you like further explanations or step-by-step derivations?

Related Questions:

1. How do you compute higher-order partial derivatives?
2. What are the applications of partial derivatives in physics and engineering?
3. How do you determine if a function is differentiable at a point?
4. What is the geometric interpretation of partial derivatives?
5. How do you use the gradient vector in optimization problems?

Tip:

When differentiating functions involving exponentials, logarithms, or trigonometric terms, always check if the chain rule, product rule, or quotient rule applies!