Differential Calculus
Limits and Continuity
Limits describe the behavior of a function as the input approaches a value.
Continuity means the limit equals the function value at that point.
$$\lim_{x \to a} f(x) = L$$
$$\text{Continuous at } x=a \iff
\begin{cases}
f(a) \text{ is defined} \\
\lim_{x \to a} f(x) \text{ exists} \\
\lim_{x \to a} f(x) = f(a)
\end{cases}
$$
Types of discontinuities: removable, jump, infinite.
Definition of the Derivative
The derivative measures instantaneous rate of change and slope of the tangent line.
$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
$$f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$$
Tangent and Normal Lines
1. Equation of the tangent line at $x=a$:
$$y - f(a) = f'(a)(x - a)$$
2. Equation of the normal line at $x=a$:
$$y - f(a) = -\frac{1}{f'(a)}(x - a)$$
Linear Approximation and Differentials — Key Formulas
$$L(x) = f(a) + f'(a)(x-a)$$
$$dy = f'(x)\, dx$$
$$\Delta y \approx dy$$
Basic Differentiation Rules
$$\frac{d}{dx}[c] = 0$$
$$\frac{d}{dx}[x^n] = nx^{n-1}$$
$$\frac{d}{dx}[c f(x)] = c f'(x)$$
$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$
Product, Quotient, and Chain Rules
$$\frac{d}{dx}(uv) = u'v + uv'$$
$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$$
$$\frac{d}{dx}[f(g(x))] = f'(g(x))\, g'(x)$$
Derivatives of Trigonometric Functions
$$\frac{d}{dx}(\sin x) = \cos x$$
$$\frac{d}{dx}(\cos x) = -\sin x$$
$$\frac{d}{dx}(\tan x) = \sec^2 x$$
$$\frac{d}{dx}(\cot x) = -\csc^2 x$$
$$\frac{d}{dx}(\sec x) = \sec x \tan x$$
$$\frac{d}{dx}(\csc x) = -\csc x \cot x$$
Derivatives of Inverse Trigonometric Functions
$$\frac{d}{dx}(\sin^{-1} u) = \frac{u'}{\sqrt{1 - u^2}}$$
$$\frac{d}{dx}(\cos^{-1} u) = \frac{-u'}{\sqrt{1 - u^2}}$$
$$\frac{d}{dx}(\tan^{-1} u) = \frac{u'}{1 + u^2}$$
$$\frac{d}{dx}(\cot^{-1} u) = \frac{-u'}{1 + u^2}$$
$$\frac{d}{dx}(\sec^{-1} u) = \frac{u'}{|u|\sqrt{u^2 - 1}}$$
$$\frac{d}{dx}(\csc^{-1} u) = \frac{-u'}{|u|\sqrt{u^2 - 1}}$$
Derivatives of Exponential and Logarithmic Functions
$$\frac{d}{dx}[e^x] = e^x$$
$$\frac{d}{dx}[\ln x] = \frac{1}{x}$$
$$\frac{d}{dx}[a^x] = a^x \ln a$$
$$\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$
Hyperbolic Functions
Certain combinations of the exponential functions $e^x$ and $e^{-x}$ occur
frequently in mathematics, science, and engineering. These are called
hyperbolic functions.
Definitions of hyperbolic functions:
$$ \sinh x = \frac{e^{x} - e^{-x}}{2} $$
$$ \cosh x = \frac{e^{x} + e^{-x}}{2} $$
$$ \tanh x = \frac{\sinh x}{\cosh x} $$
$$ coth x = \frac{\cosh x}{\sinh x} $$
$$ sech x = \frac{1}{\cosh x} $$
$$ csch x = \frac{1}{\sinh x} $$
Names are read as “hyperbolic sine,” “hyperbolic cosine,” and so on.
Fundamental Hyperbolic Identities
$$ \cosh^{2} x - \sinh^{2} x = 1 $$
$$ \tanh^{2} x + sech^{2} x = 1 $$
$$ coth^{2} x - csch^{2} x = 1 $$
$$ \sinh 2x = 2 \sinh x \cosh x $$
$$ \cosh 2x = \cosh^{2} x + \sinh^{2} x
= 1 + 2\sinh^{2} x
= 2\cosh^{2} x - 1 $$
Differentiation of Hyperbolic Functions
The derivative rules for hyperbolic functions (where $u$ is a function of $x$) are:
$$ \frac{d}{dx}(\sinh u) = \cosh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(\cosh u) = \sinh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(\tanh u) = sech^{2} u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(coth u) = -csch^{2} u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(sech u) = -sech u \tanh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(csch u) = -csch u \coth u \, \frac{du}{dx} $$
Higher Order and Implicit Differentiation
$$y' = \frac{dy}{dx}$$
$$y'' = \frac{d^2 y}{dx^2}$$
$$y^{(n)} = \frac{d^n y}{dx^n}$$
Implicit differentiation:
$$\frac{d}{dx}F(x,y) = F_x + F_y \frac{dy}{dx} = 0$$
$$\frac{dy}{dx} = -\frac{F_x}{F_y}$$
Linear Approximation and Differentials
$$dy = f'(x)\, dx$$
$$\Delta y \approx dy = f'(x)\Delta x$$
$$L(x) = f(a) + f'(a)(x-a)$$
Monotonicity and Extrema
$$f'(x) > 0 \Rightarrow \text{increasing}$$
$$f'(x) < 0 \Rightarrow \text{decreasing}$$
Concavity, Inflection Points, and the Second Derivative Test
The second derivative $f''(x)$ tells us how the graph of a function bends.
Understanding concavity helps us identify inflection points and determine whether
critical points are maxima or minima.
Concavity of the graph:
$$f''(x) > 0 \Rightarrow \text{the graph is concave up (shaped like a cup)}$$
$$f''(x) < 0 \Rightarrow \text{the graph is concave down (shaped like a hill)}$$
Inflection points:
An inflection point occurs where the graph changes concavity — from concave up to concave down or vice versa.
This usually happens where $f''(x)$ changes sign.
$$\text{If } f''(x) \text{ changes sign at } x = c,\; \text{then } c \text{ is an inflection point.}$$
The Second Derivative Test for maxima and minima:
If a point $x = c$ is a critical point (meaning $f'(c) = 0$), then the second derivative helps determine
whether it is a peak or a valley.
$$f''(c) > 0 \Rightarrow \text{local minimum at } c \quad (\text{graph curves upward})$$
$$f''(c) < 0 \Rightarrow \text{local maximum at } c \quad (\text{graph curves downward})$$
$$f''(c) = 0 \Rightarrow \text{test is inconclusive; use another method}$$
Taken together, concavity and the second derivative test provide a complete picture of how
the function behaves: where it bends, where it reaches peaks and valleys, and where the
direction of bending changes.
Optimization Problems (Maxima-Minima)
How to solve optimization problems:
- Write an equation for the quantity you want to maximize or minimize.
- Differentiate the equation and set $f'(x)=0$ to find possible maximum or minimum values.
- Check which value actually gives the largest or smallest result by comparing the outputs of $f(x)$.
These problems usually involve finding the best (maximum or minimum) value under certain conditions,
such as fixed perimeter, fixed area or volume, limited materials, or being restricted to a given path or shape.
Steps in Solving Implicit Differentiation Problems
How to differentiate equations where $y$ is not isolated:
- Differentiate both sides of the equation with respect to $x$.
- Apply the chain rule whenever you differentiate a term involving $y$, which produces $\frac{dy}{dx}$.
- Collect all terms containing $\frac{dy}{dx}$ on one side of the equation.
- Factor out $\frac{dy}{dx}$.
- Solve for $\frac{dy}{dx}$.
Implicit differentiation is used when the equation cannot easily be solved for $y$, such as circles, ellipses, or equations where $x$ and $y$ are mixed together.
Radius of Curvature
The radius of curvature measures how sharply a curve bends at a point.
1. If the curve is given as $y = f(x)$:
$$R = \frac{\left(1 + (y')^2\right)^{3/2}}{|y''|}$$
2. If the curve is given parametrically as $x(t)$ and $y(t)$:
$$R = \frac{\left(\dot{x}^2 + \dot{y}^2\right)^{3/2}}
{|\dot{x}\ddot{y} - \dot{y}\ddot{x}|}$$
3. If the curve is given in the form $x = g(y)$:
$$R = \frac{\left(1 + (x')^2\right)^{3/2}}{|x''|}$$
4. Curvature $\kappa$ (reciprocal of the radius of curvature):
$$\kappa = \frac{1}{R}$$
L'Hospital's Rule
$$\text{If } \lim_{x\to a} f(x)=0 \text{ and } \lim_{x\to a} g(x)=0$$
$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$
Mean Value Theorem
$$f \text{ continuous on } [a,b],\; f \text{ differentiable on } (a,b)$$
$$\exists c\in(a,b) \text{ such that } f'(c) = \frac{f(b)-f(a)}{b-a}$$
Rolle's Theorem: if f(a) = f(b), then there exists a number c with $f'(c)=0$.
Steps in Finding Partial Derivatives
How to compute partial derivatives of a function of two or more variables:
-
Identify which variable you are differentiating with respect to
(usually $x$, $y$, or $z$).
-
Treat all other variables as constants.
Only the chosen variable is allowed to change.
-
Differentiate the function using ordinary differentiation rules
(power rule, product rule, chain rule, etc.).
-
Write the result using partial derivative notation such as
$\frac{\partial f}{\partial x}$ or $f_x$.
-
If needed, repeat the process to find higher-order partial derivatives
like $f_{xx}$, $f_{yy}$, or mixed partials such as $f_{xy}$ and $f_{yx}$.
Notation for partial derivatives:
$$f_x = \frac{\partial f}{\partial x}$$
$$f_y = \frac{\partial f}{\partial y}$$
$$f_{xx} = \frac{\partial^2 f}{\partial x^2}$$
$$f_{xy} = \frac{\partial^2 f}{\partial x \partial y}$$
Partial derivatives appear in problems involving surfaces, optimization of functions of two variables,
and models where a quantity depends on more than one input.
