Derivative - Integration Connections and Multivariable Extensions

Antiderivatives and the Fundamental Theorem of Calculus What Is an Antiderivative? An antiderivative of a function $f$ is a function $F$ such that: $$F'(x) = f(x)$$ In other words, $F$ is a function whose derivative gives you back $f$. This is the inverse operation of differentiation. Example: If $f(x) = 3x^2$, then $F(x) = x^3$ is an antiderivative because $\frac{d}{dx}(x^3) = 3x^2$. The notation for antiderivatives uses the integral symbol: we write $F(x) = \int f(x) \, dx$ to mean "$F$ is an antiderivative of $f$." The Family of Antiderivatives Here's a crucial observation: if $F(x)$ is an antiderivative of $f(x)$, then so is $F(x) + C$ for any constant $C$. Why? Because the derivative of any constant is zero: $$\frac{d}{dx}[F(x) + C] = F'(x) + 0 = f(x)$$ This means every function has infinitely many antiderivatives—they form a family of functions that differ only by a constant. When we write the indefinite integral: $$\int f(x) \, dx = F(x) + C$$ the "$+C$" explicitly accounts for this entire family. The constant $C$ is called the constant of integration, and it's essential to include it when finding antiderivatives. Example: Both $x^3$ and $x^3 + 5$ and $x^3 - 7$ are all antiderivatives of $3x^2$, since they all have the same derivative. The Fundamental Theorem of Calculus – Part I The Fundamental Theorem of Calculus (Part I) is one of the most important theorems in mathematics. It connects differentiation and integration: $$\inta^b f(x) \, dx = F(b) - F(a)$$ where $F$ is any antiderivative of $f$ on the closed interval $[a,b]$. This theorem tells us that to compute a definite integral (which represents the signed area under the curve), we need only: Find any antiderivative $F$ of $f$ Evaluate it at the upper and lower limits Subtract: $F(b) - F(a)$ Notice that the constant $C$ in the family of antiderivatives cancels out when we compute $F(b) - F(a)$, so it doesn't matter which antiderivative we choose. Example: To find $\int0^2 3x^2 \, dx$: An antiderivative of $3x^2$ is $F(x) = x^3$ Evaluate: $F(2) - F(0) = 8 - 0 = 8$ Computing Definite Integrals via Antiderivatives The practical impact of the Fundamental Theorem is that it gives us a computational method for evaluating definite integrals. Rather than computing limits of Riemann sums (which is often difficult), we find an antiderivative and evaluate. The process is straightforward: Identify the integrand $f(x)$ and limits $a$ and $b$ Find an antiderivative $F(x)$ using integration techniques Apply the theorem: compute $F(b) - F(a)$ This is useful because many definite integrals that arise in applications (computing areas, volumes, work done by a force, etc.) can now be solved exactly rather than approximated. Multivariable Derivatives When we move beyond single-variable functions to functions of multiple variables, derivatives become more sophisticated. Instead of a single slope, we need to understand how the function changes in many different directions simultaneously. The Gradient Vector For a real-valued function $f(x1, x2, \ldots, xn)$ (a function that takes multiple inputs and produces one output), the gradient is a vector containing all of the first-order partial derivatives: $$\nabla f = \left( \frac{\partial f}{\partial x1}, \frac{\partial f}{\partial x2}, \ldots, \frac{\partial f}{\partial xn} \right)$$ The symbol $\nabla$ is called "del" or "nabla," and $\nabla f$ is read as "grad $f$" or "the gradient of $f$." Intuition: Each partial derivative $\frac{\partial f}{\partial xi}$ tells you how fast $f$ changes when you move in the $xi$ direction (while holding all other variables fixed). The gradient bundles all these rates of change into a single vector. Example: For $f(x,y) = 3x^2 + 2xy + y^2$: $$\frac{\partial f}{\partial x} = 6x + 2y$$ $$\frac{\partial f}{\partial y} = 2x + 2y$$ $$\nabla f = (6x + 2y, 2x + 2y)$$ At the point $(1, 1)$: $\nabla f(1,1) = (8, 4)$ Directional Derivative While a partial derivative tells you the rate of change in the direction of a coordinate axis, the directional derivative tells you the rate of change in any direction you choose. The directional derivative of $f$ in the direction of a unit vector $\mathbf{u}$ is: $$D{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$$ This is simply the dot product of the gradient vector with your chosen direction vector. The result is a scalar that tells you the instantaneous rate of change of $f$ as you move in the direction $\mathbf{u}$. Key insight: The directional derivative is maximized when $\mathbf{u}$ points in the same direction as $\nabla f$. This means the gradient points in the direction of steepest increase. Example: For $f(x,y) = 3x^2 + 2xy + y^2$ at point $(1,1)$ with $\nabla f = (8, 4)$: In the direction $\mathbf{u} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$: $$D{\mathbf{u}} f = (8, 4) \cdot \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) = \frac{8 + 4}{\sqrt{2}} = 6\sqrt{2}$$ The Jacobian Matrix So far we've considered functions that output a single number. What if we have a vector-valued function that outputs multiple numbers? For a vector-valued function $\mathbf{F}(x1, x2, \ldots, xn) = (f1(x1, \ldots, xn), f2(x1, \ldots, xn), \ldots, fm(x1, \ldots, xn))$, the Jacobian matrix is the matrix of all first-order partial derivatives: $$J = \begin{pmatrix} \frac{\partial f1}{\partial x1} & \frac{\partial f1}{\partial x2} & \cdots & \frac{\partial f1}{\partial xn} \\ \frac{\partial f2}{\partial x1} & \frac{\partial f2}{\partial x2} & \cdots & \frac{\partial f2}{\partial xn} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial fm}{\partial x1} & \frac{\partial fm}{\partial x2} & \cdots & \frac{\partial fm}{\partial xn} \end{pmatrix}$$ The Jacobian has one row for each output component and one column for each input variable. Each entry tells you how one output responds to changes in one input. Structure: Row $i$ contains all partial derivatives of output $fi$ with respect to each input Column $j$ contains all partial derivatives of every output with respect to input $xj$ Example: For $\mathbf{F}(x, y) = (x^2 + y, xy)$: $$J = \begin{pmatrix} 2x & 1 \\ y & x \end{pmatrix}$$ At point $(1, 2)$: $J(1,2) = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}$ Total Derivative (The Differential) The total derivative (or differential) is a generalization of the derivative from single-variable calculus. It's the best linear approximation to your function at a point. For a multivariable function, the total derivative at a point is a linear transformation that, when applied to a small change in inputs, gives you the approximate change in outputs. This linear transformation can be represented by the Jacobian matrix. Intuition: Just as a single-variable function is approximated by its tangent line, a multivariable function is approximated by its tangent hyperplane. The total derivative encodes all the information needed to define this tangent hyperplane. If $\mathbf{F}$ is differentiable at a point $\mathbf{a}$, then near that point: $$\mathbf{F}(\mathbf{a} + \mathbf{h}) \approx \mathbf{F}(\mathbf{a}) + J(\mathbf{a}) \cdot \mathbf{h}$$ where $J(\mathbf{a})$ is the Jacobian matrix at $\mathbf{a}$, and $\mathbf{h}$ is a small change in the input. Relationship Between Total and Partial Derivatives This is an important conceptual point: partial derivatives are special cases of the total derivative. The relationship: If the total derivative exists at a point, then every partial derivative exists at that point, and the partial derivatives are exactly the entries of the Jacobian matrix representing the total derivative. More precisely: The partial derivative $\frac{\partial fi}{\partial xj}$ (how output $fi$ changes with respect to input $xj$) is the $(i,j)$ entry of the Jacobian All partial derivatives existing and being continuous guarantees that the total derivative exists Why this matters: Partial derivatives alone don't tell the complete story about how a function changes. The total derivative is the complete description of how a function changes in all directions simultaneously. But practically, we can find the total derivative by computing partial derivatives and arranging them in a Jacobian matrix. Example: For $f(x,y) = x^2y$: $\frac{\partial f}{\partial x} = 2xy$ (first entry of gradient) $\frac{\partial f}{\partial y} = x^2$ (second entry of gradient) These two partial derivatives form the gradient vector, which represents the total derivative of this scalar-valued function