Partial Derivatives

The Partial Derivative #

Hold every variable but one still.

Hold every variable but one constant, and differentiate as usual

A surface has no single slope: it depends on which direction you set off walking.

Fix y and walk only in the x direction. That slice is an ordinary curve again.

That slope is the partial derivative $\frac{\partial f}{\partial x}$∂f/∂x: differentiate in x, holding y constant.

Computing Partial Derivatives #

Every ordinary rule still applies.

Every ordinary rule still applies, with the other variables treated as constants

Differentiate with respect to x: y is held constant and stays as a factor.

Differentiate with respect to y: $x^2$x² is the constant factor, and y differentiates to 1.

A term that does not contain the variable you differentiate with respect to is a constant, so it vanishes.

Higher Partial Derivatives #

Differentiate again, in either variable.

Partial derivatives can be differentiated again, in either variable

A partial derivative is itself a function of both variables, so it can be differentiated again.

Differentiating twice in x measures how the slope in x changes as you move in x.

Differentiate once in each and you measure how the slope in x changes as you move in y.

The Mixed Derivative Theorem #

x then y or y then x: the same answer.

For a well-behaved function the order of mixed partial differentiation makes no difference

Differentiate in x then y, or in y then x, and the answer comes out the same.

Try it: $2x{y}^3$2xy³ differentiated in y, and $3x^2{y}^2$3x²y² differentiated in x, both give $6x{y}^2$6xy².

It holds whenever the mixed partials are continuous, which is nearly always.

The Chain Rule for Partial Derivatives #

Both variables depend on time, so both count.

When both variables depend on time, each one contributes its own term to the change

If x and y both change with time, z changes because of both of them at once.

Add one term per variable: the partial derivative of z, times how fast that variable changes.

With only one variable this reduces to the ordinary chain rule.

The Gradient Vector #

Collecting the partial derivatives into a vector.

The two partial derivatives form a vector that points straight uphill

Put the two partial derivatives side by side and you have a vector at every point.

That gradient vector points in the steepest uphill direction from where you stand.

Its length is how steep that climb is, so the gradient carries direction and steepness.

Directional Derivatives #

Project the gradient onto your direction.

The slope in any direction is the gradient projected onto that direction

Walking sideways across a hill is gentler than walking straight up it.

Step a in x and b in y: each part is multiplied by its own partial derivative, and the two add.

The dot product of the gradient with a unit vector gives the slope in that direction.

At right angles to the gradient the slope is zero — that is why contours are level.

Tangent Planes #

The flat plane fitted at one point.

A surface has a tangent plane where a curve has a tangent line

A curve is approximated near a point by its tangent line.

A surface is approximated near a point by a flat plane resting on it.

The two partial derivatives give the tilt in each direction, and that fixes the plane.

Stationary Points of a Surface #

Flat only when both partials vanish.

A stationary point is a place where both partial derivatives are zero

Level means level in both directions at once, so both partial derivatives must be zero.

That can be the bottom of a bowl, the top of a dome — or neither of them.

A saddle is level too, yet it rises one way and falls the other: neither a maximum nor a minimum.

Lagrange Multipliers #

At the constrained best, gradients line up.

At a constrained maximum or minimum the two gradient vectors must line up

Find the smallest value of f, but only along a fixed path — a constraint.

If the two gradients pointed different ways, you could still improve f by sliding along the path.

So at the best point the two gradients must be parallel, differing only by a multiplier $\lambda$λ.

The Volume Under a Surface #

Stack the slices twice.

Integrating the slice areas a second time stacks them into the volume under a surface

Under a curve the slices made an area. Under a surface the slices make a volume.

Cut the solid at a fixed y: the face is an area under a curve, $A\left(y\right)=\int f\left(x,y\right)dx$A(y) = ∫ f(x, y) dx.

Stack the slice areas along y: $V=\int A\left(y\right)dy$V = ∫ A(y) dy — an integral of integrals, two $\int$∫ signs.

The inner integral is ordinary integration in x, and the outer one repeats it in y.