Directional Derivatives, Gradient Vectors, Tangent Planes and Differential

Directional derivatives and gradient vectors

Definition

The derivative of $f$ at $P_0(x_0,y_0)$ in the direction of the unit vector $\vec{u}=u_1\vec{i}+u_2\vec{j}$ is the number $$\left({df\over ds}\right)_{\vec{u},P_0}=\lim_{s\rightarrow 0}{f(x_0+su_1,y_0+su_2)-f(x_0,y_0)\over s},$$ provided the limit exists.

The directional derivative is also denoted as ParseError: KaTeX parse error: Got function '\vec' with no arguments as subscript at position 4: (D_\̲v̲e̲c̲{u}f)_{P_0}.

Example

Find the derivative of $$f(x,y)=x^2+xy$$ at $P_0(1,2)$ in the direction of the unit vector $\vec{u}=(1/\sqrt{2})\vec{i}+(1/\sqrt{2})\vec{j}$.

Gradient

Definition

The gradient vector (gradient) of $f(x,y)$ at a point $P_0(x_0,y_0)$ is the vector $$\nabla f ={\partial f\over \partial x}\vec{i}+{\partial f\over \partial y}\vec{j}$$ obtained by evaluating the partial derivatives of $f$ at $P_0$.

Theorem

If $f(x,y)$ is differentiable in an open region containing $P_0(x_0,y_0)$, then $$\left({df\over ds}\right)_{\vec{u},P_0}=(\nabla f)_{P_0}\cdot \vec{u},$$ the dot product of the gradient $\nabla f$ at $P_0$ and $\vec{u}$. In brief, ParseError: KaTeX parse error: Got function '\vec' with no arguments as subscript at position 3: D_\̲v̲e̲c̲{u} f=\nabla f\….

Example

Find the derivative of $$f(x,y)=x^2+xy$$ at $P_0(1,2)$ in the direction of the unit vector $\vec{u}=\vec{i}+\vec{j}$.

Properties of the directional derivative

Note that $D_{\vec{u}}f=\nabla f\cdot \vec{u}=|\nabla f|\cos(\theta)$ because $|\vec{u}|=1$.

• The function $f$ increases most rapidly when $\cos(\theta)=1$ or when $\theta=0$ and $\vec{u}$ is the direction of $\nabla f$. The direction derivative is $|\nabla f|$.

• The function $f$ decreases most rapidly in the direction of $-\nabla f$, whose directional derivative is $-|\nabla f|$.

• Any direction $\vec{u}$ orthogonal to a gradient $\nabla f\neq \vec{0}$ is a direction of zero change in $f$ because the directional derivative is 0.

Gradients and tangents to level curves

If a differentiable function $f(x, y)$ has a constant value $c$ along a smooth curve $\vec{r} = g(t)\vec{i} + h(t)\vec{j},$ at every point $(x_0, y_0)$ in the domain of a differentiable function $f(x, y)$, the gradient of $f$ is normal to the level curve through $(x_0, y_0)$.

Tanget Line to a Level Curve

Example

Find an equation for the tangent line to the ellipse $${x^2\over 4}+y^2=2$$ at the point $(-2,1)$.

Algebra rules for gradient

• Sum rule: $\nabla (f+g) = \nabla f + \nabla g$

• Difference rule: $\nabla (f-g) = \nabla f - \nabla g$

• Constant multiple rule: $\nabla (kf) = f\nabla f$, for any $k\in \R$.

• Product rule: $\nabla (fg) = f\nabla g+ g\nabla f$

• Quotient rule: $\nabla \left({f\over g}\right) = {g\nabla f-f\nabla g\over g^2}$, when $g\neq 0$.

Functions of Three Variables

Find the derivative of $f(x,y,z)=x^3-xy^2-z$ at $P_0(1,1,0)$ in the direction of $\vec{v}=2\vec{i}-3\vec{j}+6\vec{k}$. In what directions do $f$ change most rapidly at $P_0$, and what is the rate of change in these directions?

Chain rule for paths

Find the derivative along the path $${d\over dt}f(\vec{r}(t))=\nabla f(\vec{r}(t))\cdot \vec{r}'(t)).$$

Tangent Planes and Differentials

Chain rule for paths in the space

Consider a parameterized curve $\vec{r}(t)$ such that $$c=f(\vec{r}(t))=f(x(t),y(t),z(t)).$$ Taking the derivative with respect to $t$ on both sides, we have $$0={d\over dt}f(\vec{r}(t))=\nabla f(\vec{r}(t))\cdot \vec{r}'(t)$$

• tangent line at the point $P_0$ on the curve $\vec{r}(t)$: the line through $P_0$ in the direction of $\vec{r}'$.

• tangent line at the point $P_0$ on the curve $\vec{r}(t)$ such that $f(\vec{r}(t))=c$: a line through $P_0$ orthogonal to $\nabla f$.

Tangent planes and normal lines

Definition

• tangent plane to the level surface $f(x,y,z)=c$ at the point $P_0(x_0,y_0,z_0)$: the plane through $P_0$ normal to $\nabla f|_{P_0}$.

• normal line of the surface at $P_0$: the line through $P_0$ parallel to $\nabla f|_{P_0}$.

Tangent plane to $f(x,y,z)=c$ at $P_0(x_0,y_0,z_0)$ $$f_x(P_0)(x - x_0) + f_y(P_0)(y - y_0) + f_z(P_0)(z - z_0)= 0$$ Normal line to $f(x,y,z)=c$ at $P_0(x_0,y_0,z_0)$ $$x = x_0 + f_x(P_0)t, y = y_0 + f_y(P_0)t, z = z_0 + f_z(P_0)t$$

Example

Find the tangent plane and normal line of the level surface $$f(x,y,z)=x^2+y^2+z-9=0$$ at the point $P_0(1,2,4)$.

Plane tangent to a surface $z=f(x,y)$ at $(x_0,y_0,f(x_0,y_0))$

The plane tangent to the surface $z = ƒ(x, y)$ of a differentiable function ƒ at the point $P_0(x_0 , y_0 , z_0) = (x_0 , y_0 , ƒ(x_0 , y_0))$ is $$ƒ_x(x_0 , y_0)(x - x_0) + ƒ_y(x_0 , y_0)(y - y_0) - (z - z_0) = 0.$$

Example

Find the plane tangent to the surface $z=x\cos y-ye^x$ at $(0,0,0)$.

Find the parametric equations for the line tangent to an ellipse $E$

meet in an ellipse $E$.

Estimating the change in $f$ in a direction $\vec{u}$

To estimates the change in the function value of $f$ when we move a small distance $ds$ from a point $P_0$ in the direction $\vec{u}$, we use the formula $$df= (\nabla f|_{P_0}\cdot\vec{u})ds$$

Linearize a function of TWO variables

The linearization of $f(x,y)$ at $(x_0,y_0)$ where $f$ is differentiable is the function $$L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).$$ The approximation $$f(x,y)\approx L(x,y)$$ is the standard linear approximation of $f$ at $(x_0,y_0)$.

Example

Find the linearization of $$f(x,y)=x^2-xy+{1\over 2}y^2+3$$ at $(3,2)$.

The error $f(x,y)-L(x,y)$ in the standard linear approximation

$f(x,y)-L(x,y)$ Definition If we move from $(x_0, y_0)$ to $(x_0 + dx, y_0 + dy)$ nearby, the resulting change $$df = f_x(x_0, y_0) dx + f_y(x_0, y_0) dy$$ in the linearization of $f$ is called the total differential of $f$.

Example

A cylindrical can is designed to have a radius of 1 cm and a height of 5 cm, but the radius and height are off by the amounts $dr$ = +0.03 and $dh$ = -0.1. Estimate the resulting absolute change in the volume of the can.

The error $f(x,y)-L(x,y)$ in the standard linear approximation

If $f$ has continuous first and second partial derivatives throughout an open set containing a rectangle $R$ centered at $(x_0,y_0)$ and if $M$ is any upper bound for the values of $|f_{xx}|$, $|f_{xy}|$, and $|f_{yy}|$ on $R$, then the error $E(x,y)=f(x,y)-L(x,y)$ incurred in replacing $f(x,y)$ on $R$ by its linearization $$L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$$ satisfies the inequality $$|E(x,y)|\leq {1\over2}M(|x-x_0|+|y-y_0|)^2.$$

Definition

If we move from $(x_0, y_0)$ to $(x_0 + dx, y_0 + dy)$ nearby, the resulting change $$df = f_x(x_0, y_0) dx + f_y(x_0, y_0) dy$$ in the linearization of $f$ is called the total differential of $f$.

Example

A cylindrical can is designed to have a radius of 1 cm and a height of 5 cm, but the radius and height are off by the amounts $dr$ = +0.03 and $dh$ = -0.1. Estimate the resulting absolute change in the volume of the can.

Functions of more than two variables

• The linearization of $f(x, y, z)$ at $P_0(x_0, y_0, z_0)$ is $$L(x, y, z) = f(P_0) + f_x(P_0)(x - x_0) + f_y(P_0)(y - y_0) + f_z(P_0)(z - z_0).$$

• Suppose that $R$ is a closed rectangular solid centered at $P_0$ and lying in an open region where the second partial derivatives of $f$ are continuous. Suppose also that $|f_{xx}|$, $|f_{yy}|$, $|f_{zz}|$, $|f_{xy}|$, $|f_{xz}|$, and $|f_{yz}|$ are all less than or equal to $M$ throughout $R$. Then the error $$|E(x, y, z)| = |f(x, y, z) - L(x, y, z)|\leq {M\over 2}( |x - x_0| +|y - y_0| + |z - z_0|)^2.$$

• If the second partial derivatives of $f$ are continuous and if $x$, $y$, and $z$ change from $x_0$, $y_0$, and $z_0$ by small amounts $dx$, $dy$, and $dz$, the total differential $$df = f_x(P_0) dx + f_y(P_0) dy + f_z(P_0) dz$$ gives a good approximation of the resulting change in $f$.