# Single-Variable Calculus Derivative

# Directional Derivative

## Understanding this

# Partial Derivative

## Computing Directional Derivatives

# Examples: Partial Derivatives

# Examples: Directional Derivatives

David Witten

Start by looking at the exponent.
$$=e^{x^2 + cos(xy)}\frac{\partial}{\partial x}\left(x^2 + cos(xy)\right)$$
$$=e^{x^2 + cos(xy)}\left(2x - ysin(xy)\right)$$

First, what are the partials?
$$\dfrac{\partial f}{\partial x} = 2x - 5y$$
$$\dfrac{\partial f}{\partial y} = -5x$$
Now, how can we turn the vector into a unit vector? Divide by its magnitude.
$$<3, 4>*\frac{1}{\sqrt{3^2 + 4^2}} = <\frac{3}{5},\frac{4}{5}>$$
So, the directional derivative equals
$$\frac{3}{5}\left(2x - 5y\right) +\frac{2}{5}\left(-5x\right)$$
$$=\dfrac{-4x}{5} - 3y$$

David Witten