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:
$$ \frac{d}{dx}(\sinh u) = \cosh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(\cosh u) = \sinh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(\tanh u) = sech^{2} u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(coth u) = -csch^{2} u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(sech u) = -sech u \tanh u \, \frac{du}{dx} $$
$$ \frac{d}{dx}(csch u) = -csch u \coth u \, \frac{du}{dx} $$
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.
Related Rates/Time Rates Problems
How to solve problems involving changing quantities:
Identify all quantities that are changing and assign variables to them.
Write an equation that relates these variables (from geometry, physics, or the problem setup).
Differentiate the equation with respect to time using the chain rule. This produces rates like $\frac{dx}{dt}$ or $\frac{dy}{dt}$.
Substitute all known values, including any given rates.
Solve for the unknown rate.
These problems often describe how one measurement changes while another depends on it, like water rising in a tank, a shadow growing, or two moving objects getting closer or farther apart.
Radius of Curvature
The radius of curvature measures how sharply a curve bends at a point.
$$f \text{ continuous on } [a,b],\; f \text{ differentiable on } (a,b)$$
$$\exists c\in(a,b) \text{ such that } f'(c) = \frac{f(b)-f(a)}{b-a}$$
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.
