Lecture 14: Review of Differentiability, Tangent Planes, and Introduction to the Chain Rule

Review of Differentiability and Tangent Planes (from Lecture 13)

• One-Variable Case (1D):

  • The derivative $(f'(a))$ is the slope of the tangent line to the curve $y=f(x)$ at $x=a$.

  • Approximated by slopes of secant lines, as the difference $h \to 0$.

  • Limit definition: $(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h})$.

  • Rearranged form (linearization with error term): $(f(a+h) = f(a) + f'(a)h + E(h)h)$ where $E(h) \to 0$ as $h \to 0$. This implies the error goes to zero quickly.

  • Linearization: By re-indexing and setting $x = a+h$, we get the equation of the tangent line: $(L(x) = f(a) + f'(a)(x-a))$. This is the first-degree Taylor polynomial centered at $a$.

• Two-Variable Case (2D):

  • An analogous multi-variable limit definition exists (e.g., $\lim_{(\Delta x, \Delta y) \to (0,0)} \frac{f(a+\Delta x, b+\Delta y) - f(a,b)}{\sqrt{(\Delta x)^2 + (\Delta y)^2}}$ and more complex algebraic manipulations).

  • Differentiability Definition: A function $f(x,y)$ is differentiable at $(a,b)$ if it is well approximated by its tangent plane at $(a,b)$. This is similar to the 1D case where a function is differentiable if well approximated by its tangent line.

  • Geometric Understanding of Partial Derivatives:

    • To understand partial derivatives geometrically, one slices the surface $z = f(x,y)$ with planes of fixed $x$ or $y$ values (e.g., $y = y<em>0$ or $x = x</em>0$).

    • These slices create curves on the surface. When viewed as one-variable functions, the slope of these curves is the partial derivative at that point.

    • The tangent vectors to these curves at a given point effectively span the tangent plane at that point (forming an affine space).

  • Tangent Plane as Linearization: The tangent plane is the two-variable analog of the tangent line and represents the linearization of the function at that point.

  • Differentiability Condition (easier to check): A function $f(x,y)$ is differentiable at $(a,b)$ if its partial derivatives, $f<em>x$ and $f</em>y$, exist and are continuous in a region containing $(a,b)$. This is a sufficient condition.

Differentials and Total Differential

• Concept Definition: Differentials ($dx$, $dy$) represent the linearized change in a variable, while increments ($\Delta x$, $\Delta y$) represent the actual change.

• 1D Example:

  • For the independent variable: $\Delta x = dx$.

  • For the dependent variable: $\Delta y$ (actual change in $f(x)$) vs. $dy$ (linearized change along the tangent line). The formula for the linearized change is $dy = f'(x)dx$.

• 2D Case:

  • For independent variables: $\Delta x = dx$ and $\Delta y = dy$.

  • For the dependent variable ($Z=f(x,y)$), the total differential ($dZ$) is the linearized change in $Z$ (change along the tangent plane).

  • Total Differential Formula: $(dZ = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy)$.

  • Geometric Interpretation of $dZ$: It represents how much the $Z$ coordinate changes as one moves between two points on the tangent plane, given changes $dx$ and $dy$ in the $X$ and $Y$ directions respectively.

  • Relationship to Differentiability Definition in 2D: The complicated definition of 2D differentiability, $(f(a+\Delta x, b+\Delta y) = f(a,b) + \frac{\partial f}{\partial x}(a,b)\Delta x + \frac{\partial f}{\partial y}(a,b)\Delta y + E<em>1(\Delta x)\Delta x + E</em>2(\Delta y)\Delta y)$ where $E<em>1, E</em>2 \to 0$ as $(\Delta x, \Delta y) \to (0,0)$, can be understood as:

    • The terms $\frac{\partial f}{\partial x}(a,b)\Delta x + \frac{\partial f}{\partial y}(a,b)\Delta y$ are precisely the total differential $dZ$. They represent the change along the tangent plane.

    • The terms $E<em>1(\Delta x)\Delta x + E</em>2(\Delta y)\Delta y$ are error terms ($\Delta Z - dZ$) that quantify the deviation between the actual surface and its tangent plane approximation.

• Practical Application: Error Propagation

  • Differentials can be used to estimate how uncertainties in input measurements propagate to the uncertainty of a calculated output quantity (e.g., in science labs).

  • Example: Volume of a Cone

    • Formula: $V = \frac{1}{3} \pi r^2 h$.

    • Measurements: radius $r = 10$ cm, height $h = 25$ cm.

    • Measurement errors: $\Delta r = dr = \pm 0.1$ cm, $\Delta h = dh = \pm 0.1$ cm.

    • Goal: Estimate the error in the volume ($dV$).

    • Apply total differential formula: $(dV = \frac{\partial V}{\partial r} dr + \frac{\partial V}{\partial h} dh)$.

    • Calculate partial derivatives:

      • $\frac{\partial V}{\partial r} = \frac{2}{3} \pi r h$

      • $\frac{\partial V}{\partial h} = \frac{1}{3} \pi r^2$

    • Plug in values: $(dV = (\frac{2}{3}\pi (10)(25))(0.1) + (\frac{1}{3}\pi (10)^2)(0.1))$ (values for $dr$ and $dh$ are bounds of error, so $\pm$ is often implied).

    • The calculated volume is $V = \pi (10)^2 (25) / 3$ plus or minus $dV$. This provides an estimate of the error using a linear approximation.

Chain Rule

• Motivation: To understand the rate of change of a function in terms of an indirect variable by compounding the rates of change of direct variables.

• 1D Recall: If $y = f(x)$ and $x = g(t)$, then $y$ is a differentiable function of $t$, and its derivative is given by the chain rule: $(\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt})$.

• Case 1: Two Intermediate Variables, One Independent Variable

  • Let $Z = f(x,y)$ where $x = x(t)$ and $y = y(t)$.

  • The chain rule for this case is: $(\frac{dZ}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt})$.

  • This formula can be proven using limits of difference quotients, similar to the 1D case (though skipped in class).

  • Analogy to Total Differential: It resembles the total differential formula ($dZ = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$), where effectively $dx$ and $dy$ are replaced by $\frac{dx}{dt}dt$ and $\frac{dy}{dt}dt$ and then divided by $dt$.

  • Example 1: Find $dZ/dt$ for $Z = x^2y + 3xy^4$, with $x = \sin(2t)$ and $y = \cos(t)$ when $t=0$.

    • Calculate partial derivatives of $Z$: $\frac{\partial Z}{\partial x} = 2xy + 3y^4$ and $\frac{\partial Z}{\partial y} = x^2 + 12xy^3$.

    • Calculate derivatives of $x$ and $y$ with respect to $t$: $\frac{dx}{dt} = 2\cos(2t)$ and $\frac{dy}{dt} = -\sin(t)$.

    • Evaluate $x(0)=0$ and $y(0)=1$.

    • Substitute into the chain rule formula: $\frac{dZ}{dt} = (2(0)(1) + 3(1)^4)(2\cos(0)) + ((0)^2 + 12(0)(1)^3)(-\sin(0))$

    • $= (3)(2) + (0)(0) = 6$.

  • Geometric Understanding: The path $(x(t), y(t))$ in the XY-plane creates a curve on the surface $Z=f(x,y)$. The derivative $dZ/dt$ represents the slope of the tangent vector to this curve on the surface.

  • Applied Example: Ideal Gas Law: Let $P = f(T,V)$ from $PV = NRT$, assuming $NR = 8.31$, so $P = \frac{8.31T}{V}$. Find the rate pressure changes ($dP/dt$) given rates of change for temperature ($dT/dt$) and volume ($dV/dt$).

    • Chain Rule Formula: $(\frac{dP}{dt} = \frac{\partial P}{\partial T} \frac{dT}{dt} + \frac{\partial P}{\partial V} \frac{dV}{dt})$.

    • Calculate partial derivatives:

      • $\frac{\partial P}{\partial T} = \frac{8.31}{V}$

      • $\frac{\partial P}{\partial V} = -\frac{8.31T}{V^2}$

    • Plug in given values (e.g., $V=100$, $T=300$, $dT/dt=0.1$, $dV/dt=0.1$) into the formula.

• Case 2: Two Intermediate Variables, Two Independent Variables

  • Let $Z = f(x,y)$ where $x = x(s,t)$ and $y = y(s,t)$.

  • This generalizes from Case 1 by treating one of the independent variables ($s$ or $t$) as a constant when differentiating with respect to the other.

  • Formulas:

    • $\frac{\partial Z}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}$ (treating $t$ as constant)

    • $\frac{\partial Z}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}$ (treating $s$ as constant)

  • Terminology: $Z$ is the output/dependent variable; $x,y$ are intermediate variables; $s,t$ are independent variables.

• General Case: Many Intermediate and Independent Variables

  • Let $U = f(x<em>1, x</em>2, \dots, x<em>n)$, where each intermediate variable $x</em>i$ is a function of multiple independent variables $t<em>1, t</em>2, \dots, t<em>m$ (i.e., $x</em>i = x<em>i(t</em>1, \dots, t_m)$).

  • The chain rule for any partial derivative of $U$ with respect to an independent variable $t_j$ is:

    • $(\frac{\partial U}{\partial t<em>j} = \sum</em>{i=1}^{n} \frac{\partial U}{\partial x<em>i} \frac{\partial x</em>i}{\partial t_j})$.

  • This formula sums over all intermediate variables ($x_i$) and covers all previous cases.

  • Terminology: $U$ is the output/dependent variable; $x<em>1, \dots, x</em>n$ are intermediate variables; $t<em>1, \dots, t</em>m$ are independent variables.