Welcome to multivariable calculus derivatives! In single-variable calculus, we learned that the derivative f prime of x gives us the slope of the tangent line at any point. For example, if f of x equals x squared, then f prime of x equals 2x. But what happens when our function depends on multiple variables, like f of x comma y equals x squared plus y squared? This is where multivariable derivatives come into play.
When we have a function of two variables like f of x comma y equals x squared plus y squared, its graph is no longer a curve but a three-dimensional surface. At any point on this surface, we can ask two important questions: How steep is the surface if we move in the x-direction? And how steep is the surface if we move in the y-direction? These questions lead us to partial derivatives. The partial derivative with respect to x is 2x, and the partial derivative with respect to y is 2y.
The gradient vector combines all partial derivatives into a single, powerful tool. For a function f of x comma y, the gradient is written as nabla f and equals the vector containing partial f partial x comma partial f partial y. For our example f of x comma y equals x squared plus y squared, the gradient is the vector 2x comma 2y. The gradient has two key properties: it points in the direction of steepest increase of the function, and its magnitude gives the maximum rate of change at that point.
The directional derivative extends our concept even further. While partial derivatives tell us the rate of change in the x or y directions, the directional derivative tells us the rate of change in ANY direction. The formula is D sub u of f equals the gradient of f dot product with u, where u is a unit vector pointing in the desired direction. For example, at point one comma one, if we want the rate of change in the direction one comma one, we first normalize this to get the unit vector, then take the dot product with the gradient.
To summarize what we have learned about multivariable calculus derivatives: Partial derivatives measure the rate of change in coordinate directions, holding other variables constant. The gradient vector combines all partial derivatives into a single vector that points toward the steepest increase. Directional derivatives extend this to find the rate of change in any direction we choose. These concepts form the foundation for optimization, vector calculus, and many applications in physics and engineering.