Partial derivatives, gradient, differentiability and the tangent plane
Partial derivatives and the gradient, why they are not enough, differentiability and the tangent plane, directional derivative and steepest ascent, the chain rule and the Hessian.
In one variable the derivative tells how fast a function grows. In several variables the question becomes: how fast does it grow moving in a given direction? The answer leads to partial derivatives, the gradient, the tangent plane and differentiability, which is the right notion for saying that a function is “smooth”.
Partial derivatives
The partial derivative with respect to 
is computed by holding 
fixed and differentiating as in Analysis 1:
![\[ \frac{\partial f}{\partial x}(x_0, y_0) = \lim_{h\to 0}\frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-750559939aeb63a2bc1f90a2bb43af16_l3.png "Rendered by QuickLaTeX.com")
Geometrically it is the slope of the curve obtained by cutting the graph with the plane 
. Example: 
has 
and 
.
The vector of partial derivatives is the gradient:
![\[ \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-e1ef4b25455de0a2cfc9c72477c84826_l3.png "Rendered by QuickLaTeX.com")
Partial derivatives are not enough
In one variable, differentiable implies continuous. In several variables it does not: the function 
, extended by 0 at the origin, has both partial derivatives zero at 
(along the axes it is always 0), but it is not even continuous there. Partial derivatives look at only two directions, while the behaviour around a point depends on all of them.
Differentiability and the tangent plane
is differentiable at 
if it can be approximated by a linear function with an error that goes to zero faster than the distance:
![\[ f(\mathbf x_0 + \mathbf h) = f(\mathbf x_0) + \nabla f(\mathbf x_0)\cdot\mathbf h + o(\|\mathbf h\|) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-3bb8ce241511b8fb29b622320b5cbda6_l3.png "Rendered by QuickLaTeX.com")
Geometrically: near the point the graph cannot be told apart from a plane, the tangent plane
![\[ z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-940fa96d693b764d75735fd69d80533e_l3.png "Rendered by QuickLaTeX.com")
Differentiable implies continuous and having every directional derivative. The practical test: if the partial derivatives exist and are continuous near the point, the function is differentiable (total differential theorem). All functions built from elementary operations are, where the derivatives are continuous.
Example. The tangent plane to 
at 
: 
, 
, so 
.
Directional derivative
The derivative in the direction of the unit vector 
is
![\[ \frac{\partial f}{\partial\mathbf v}(\mathbf x_0) = \lim_{t\to 0}\frac{f(\mathbf x_0 + t\mathbf v) - f(\mathbf x_0)}{t} \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-e4b2351b6420caaa699f7ff3ce9cc9c9_l3.png "Rendered by QuickLaTeX.com")
and, if is differentiable, the gradient formula holds:
![\[ \frac{\partial f}{\partial\mathbf v} = \nabla f\cdot\mathbf v = |\nabla f|\cos\theta \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-48c4f67e9d8e91bab0f10f11cec8f87f_l3.png "Rendered by QuickLaTeX.com")
Proof and consequences
From the definition of differentiability with 
: 
; dividing by 
and taking the limit gives the formula. From the cosine three key facts follow:
- the directional derivative is largest in the direction of the gradient (
) and equals 
: the gradient points in the direction of steepest ascent; - it is smallest,
, in the opposite direction; - it is zero in directions perpendicular to the gradient.
and the gradient at a few points: it is always perpendicular to the level curve through the point and points towards increasing values. Where the curves are closer together the gradient is longer.Along a level curve is constant, so the derivative in the tangent direction is zero: the gradient is perpendicular to the level curves. That is why electric field lines 
cross equipotential surfaces at right angles, and why water runs down the steepest slope, perpendicular to the contour lines.
The chain rule
If 
and 
describe a curve, the composite function 
has derivative
![\[ g'(t) = f_x\,x'(t) + f_y\,y'(t) = \nabla f\cdot\mathbf r'(t) \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-3a68bab1d0fc13e09f854f3c371744c2_l3.png "Rendered by QuickLaTeX.com")
Example. The temperature on a plate is 
°C (with 
in metres) and an ant moves along 
metres, with in seconds. At time 
, 
and 
: the ant feels the temperature drop by 
°C per second.
More generally, for 
the derivative is the Jacobian matrix, with the partial derivatives of each component along the rows, and the chain rule says that the Jacobian of a composition is the product of the Jacobians. It is the bridge between Analysis 2 and Geometry and Algebra: differentiating means approximating with a linear map.
Second derivatives
Differentiating again gives the second derivatives 
, collected in the Hessian matrix. Schwarz’s theorem guarantees that, if the mixed second derivatives are continuous, the order does not matter: 
, and the Hessian is symmetric. We will use it in the next lesson to classify maxima, minima and saddle points.
