Derivative of Cosine Function

If $f(x) = \cos(x)$, then $f'(x) = -\sin(x)$.

Meaning of the Exponential (e)

$e \approx 2.71828182\dots$ is defined as the limit of $(1 + \frac{1}{n})^n$ as $n \to \infty$.

This arises in compound interest: if you earn 100% interest per year, compounding more frequently yields more money, approaching $e$ times your initial investment.

Compounding once: $(1+1)^1 = 2$
Compounding twice (50% every 6 months): $(1+\frac{1}{2})^2 = 2.25$
Compounding three times (33.3% every 4 months): $(1+\frac{1}{3})^3 \approx 2.37$
Compounding infinitely often: $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$

Derivative of $e^x$

A unique property of $e^x$ is that its derivative is itself:

$f(x) = e^x \Rightarrow f'(x) = e^x$

This means the slope of $e^x$ at any point $x$ is equal to the value of $e^x$ at that point.

For $f(x) = a^x$, its derivative is $f'(x) = \ln(a)a^x$.

Derivative of $\ln(x)$

The natural logarithm function, $\ln(x)$, is the inverse of $e^x$.

If $f(x) = e^x$, then $f^{-1}(y) = \ln(y)$.

Using the inverse function derivative rule $g'(y) = \frac{1}{f'(f^{-1}(y))}$:

$f'(x) = e^x$

So, $\frac{d}{dy}\ln(y) = \frac{1}{e^{\ln(y)}} = \frac{1}{y}$.

Therefore, if $f(x) = \ln(x)$, then $f'(x) = \frac{1}{x}$.

Existence of the Derivative

Differentiable Functions

A function is differentiable at a point if its derivative exists at that point. Geometrically, this means there is a unique tangent line at that point.

A function is differentiable over an interval if it is differentiable at every point in the interval.

Non-Differentiable Functions

A function is not differentiable at points where:

Corners or Cusps: The slope changes abruptly, leading to different left and right derivatives (e.g., $f(x) = |x|$ at $x=0$).
Jump Discontinuities: The function has a sudden break or jump (e.g., piecewise functions with gaps).
Vertical Tangents: The tangent line is vertical, meaning the slope is infinite (e.g., $f(x) = x^{1/3}$ at $x=0$).

Properties of the Derivative: Multiplication by Scalars

If $f(x) = c \cdot g(x)$, where $c$ is a constant, then the derivative is:

$f'(x) = c \cdot g'(x)$

Multiplying a function by a scalar $c$ also multiplies its slope/derivative by $c$.

Properties of the Derivative: The Sum Rule

If $f(x) = g(x) + h(x)$, then the derivative is:

$f'(x) = g'(x) + h'(x)$

The derivative of a sum of functions is the sum of their derivatives.

Example: If a boat moves at velocity $v_B$ and a child walks on it at $v_C$ in the same direction, the child's total velocity relative to the earth is $v_{Total} = v_B + v_C$. If $v_B = \frac{dx_B}{dt}$ and $v_C = \frac{dx_C}{dt}$, then $v_{Total} = \frac{dx_B}{dt} + \frac{dx_C}{dt}$.

Properties of the Derivative: The Product Rule

If $f(t) = g(t)h(t)$, then the derivative is:

$f'(t) = g'(t)h(t) + g(t)h'(t)$

This can be visualized by considering the change in area of a rectangle with sides $g(t)$ and $h(t)$ over time.

Properties of the Derivative: The Chain Rule

If $f(t) = g(h(t))$, then the derivative is:

$f'(t) = g'(h(t)) \cdot h'(t)$

In Leibniz notation: $\frac{df}{dt} = \frac{dg}{dh} \cdot \frac{dh}{dt}$

Example: If temperature ($T$) changes with height ($h$), and height ($h$) changes with time ($t$), then the rate of change of temperature with respect to time is $\frac{dT}{dt} = \frac{dT}{dh} \cdot \frac{dh}{dt}$.

Introduction to Optimization

Optimization involves finding the inputs that result in the minimum or maximum o