1.7 Partial Derivatives

The use of calculators or other technologies are strictly prohibited for this assignment.

1 Intro to Partial Derivatives

Consider \(f(x) = x^2 y + \sin(y)\).

  1. Find \(\frac{\partial f}{\partial x}\).

  2. What is the partial derivative of \(f\) with respect to \(x\), evaluated at (1, 1)?

  3. Find \(\frac{\partial f}{\partial y}\).

  4. What is the partial derivative of \(f\) with respect to \(x\), evaluated at (1, 0)?

2 More Partial Derivatives

Consider \(f(x) = x^{3/4}y^{1/4}\).

  1. Find \(\frac{\partial f}{\partial x}\).

  2. Find \(\frac{\partial f}{\partial y}\).

  3. Simplify \(\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}\).

Consider \(f(x) = \frac{3}{4} \log(x) + \frac{1}{4} \log(y)\)

  1. Find \(\frac{\partial f}{\partial x}\).

  2. Find \(\frac{\partial f}{\partial y}\).

  3. Find \(\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}\).