Partial derivatives and the gradient
Partial derivatives of functions of two variables, the gradient at a point, the equation of the tangent plane and stationary points.
How to solve it
The partial derivative with respect to $x$ is an ordinary derivative in which $y$ is treated as a constant. The two partial derivatives together form the gradient, a vector pointing in the direction of the fastest growth.
Step by step
- $\frac{\partial f}{\partial x}$: differentiate in $x$, treating $y$ as a number. Terms without $x$ give zero.
- The gradient at a point: find both partial derivatives and substitute the coordinates.
- The tangent plane at $(x_0,\ y_0)$: $z = f(x_0, y_0) + f_x(x - x_0) + f_y(y - y_0)$.
- A stationary point: both partial derivatives are zero — solve the system.
Common mistakes
- Differentiating $y$ as a variable when taking $\frac{\partial}{\partial x}$: the term $y^2$ gives $0$, and $4xy$ gives $4y$.
- Forgetting the chain rule: $\frac{\partial}{\partial x}\sin(2x - y) = 2\cos(2x - y)$.
- Putting expressions instead of numbers into the tangent plane: substitute the point first.
- Setting only one partial derivative to zero when looking for a stationary point.