Partial derivatives and gradients

Change in more than one direction

Most quantities in finance depend on several inputs at once. The value of a portfolio depends on every asset price; an option's price depends on the stock price, the volatility, the rate and the time. A partial derivative asks how the function changes when you move one input and hold all the others fixed.

Collect all the partial derivatives in a vector and you get the gradient $\nabla f$. It points in the direction of steepest increase, its length is the steepest rate of increase, and a small move $\Delta\mathbf{x}$ changes the function by about $\nabla f\cdot\Delta\mathbf{x}$. Gradient descent, which trains most machine-learning models, simply steps against the gradient.

• Partial derivative $f_x$: the rate of change in $x$ with the other inputs held fixed.
• Gradient $\nabla f$: the vector of all partial derivatives.
• Steepest ascent: the direction of $\nabla f$; its length $\|\nabla f\|$ is the steepest rate.
• First-order change: $\Delta f\approx\nabla f\cdot\Delta\mathbf{x}$ for a small move.

Lab · what's graded

• Partial in x15 pts
• Partial in y15 pts
• The gradient at a point15 pts
• The steepest rate15 pts
• What the gradient points to10 pts
• Computing a gradient10 pts
• Holding the others fixed20 pts