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Partial derivatives measure the change with respect to one variable while holding the others fixed. The gradient collects these rates into a vector.
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Formula
Units: each component has units of f divided by the corresponding coordinate unit.
| Symbol | Meaning | Unit |
|---|---|---|
| Scalar field or function | same unit as f | |
| Independent coordinate | respective coordinate unit | |
| Rate of change in coordinate direction i | unit of f / unit of xi |
\nabla f = (\partial f/\partial x_1,\ldots,\partial f/\partial x_n)A partial derivative is formed by holding all other independent variables fixed. The gradient points in the direction of the steepest local increase; its magnitude is the maximum directional derivative. It is perpendicular to a level surface.
Applies to differentiable scalar functions; the gradient may fail to exist at nondifferentiable points.
Each component is an ordinary derivative with the other variables held fixed.
D_u f=∇f·u
Applies to differentiable scalar functions; the gradient may fail to exist at nondifferentiable points.
For f(x,y)=x2+xy+y2, find ∇f at (1,2).
For f(x,y)=x2+xy+y2, ∇f=(2x+y,x+2y).
At (1,2), ∇f=(4,5).
In direction u=(3/5,4/5), f initially increases at ∇f·u=32/5=6.4 per unit length.
For the small displacement (0.002,0.004), linearization predicts Δf≈0.028; direct calculation gives 0.028028.
Differentiate the other variables as well.
When differentiating with respect to x, hold y constant; substitute the point only afterward.
Formula cluster
The gradient turns several partial rates into the direction and magnitude of local increase.
For f(x,y)=x2+xy+y2, ∇f=(2x+y,x+2y). At (1,2), ∇f=(4,5). In direction u=(3/5,4/5), f initially increases at ∇f·u=44/5=8.8 per unit length. Check: f(1.002,2.004)−f(1,2)≈0.044, consistent with 8.8·0.005.
Local rates of change, tangent planes, optimization and potential fields in physics and engineering.
Here is how to work through a typical Partial derivatives and gradient (∇f = (∂f/∂x₁, …, ∂f/∂xₙ)) task step by step:
Task
For f(x,y)=x2+xy+y2, find ∇f at (1,2).
Solution path
For f(x,y)=x2+xy+y2, ∇f=(2x+y,x+2y).
At (1,2), ∇f=(4,5).
In direction u=(3/5,4/5), f initially increases at ∇f·u=32/5=6.4 per unit length.
For the small displacement (0.002,0.004), linearization predicts Δf≈0.028; direct calculation gives 0.028028.
When forming ∂f/∂x, y is not varied. For f=x2y+3y2, differentiation with respect to x gives 2xy; 3y2 is constant and differentiates to zero. Substitute a point only after computing the partial derivatives. For small changes, the tangent plane gives a linear approximation. The error usually shrinks with the step because higher-order terms contribute less. For large changes, the gradient alone is insufficient; evaluate the function again or use a fuller model. Compare the approximation with a direct function evaluation when the step is not very small. This shows whether curvature already changes the linear prediction appreciably.
For a unit vector u, the directional derivative is Du f=∇f·u=|∇f| cos θ. It is largest when u is parallel to the gradient, so cos θ=1. The opposite direction gives the steepest decrease; perpendicular directions have zero first-order change. Make sure u is normalized when you want change per unit distance. Without normalization, the dot product also scales with the length of the direction vector. The largest directional derivative equals the gradient magnitude, while directions along a level surface keep the function locally constant.
If f is temperature in °C and x,y are lengths in metres, each gradient component has units °C/m. Coordinates with different units can give components with different units. The vector magnitude therefore has a physical interpretation only after the coordinates are sensibly scaled. A unit check prevents mixing temperature changes with distances. Under a coordinate transformation, components change by the chain rule, so values cannot be compared independently of the chosen scale. Applications often normalize coordinates before interpreting gradient magnitudes or optimization steps.
Along a level curve, f stays constant, so motion tangent to the curve has directional derivative zero. Since Du f=∇f·u, the gradient is perpendicular to a tangent. This helps determine tangent and normal directions. For example, a tangent to f(x,y)=c is orthogonal to ∇f, provided the gradient is nonzero. A normal vector to the curve is therefore immediately available. If ∇f=0, this rule gives no preferred direction; higher derivatives or a local sketch of the level curve may be needed. For f(x,y)=x2+y2, level sets are circles and ∇f=(2x,2y) points radially outward except at the origin.
The gradient requires all partial derivatives at the point. At jumps, cusps or points lacking partial derivatives, it may not exist. Even existing partial derivatives alone do not guarantee differentiability, though standard smooth functions usually meet the stronger condition. Differentiability means more than the existence of separate partial derivatives. It requires one common linear approximation for changes in every direction. For piecewise functions, check the point separately; existing partial derivatives alone do not guarantee a well-defined gradient. If derivatives fail at a cusp, a unique local direction of slope is not guaranteed. Check limits along different directions instead of forcing a gradient interpretation.
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