# Derivatives

## Tangent line and rate of change

A *tangent line* touches a curve exactly once.

```{image} ./_images/tangent.png
:alt: Graph of tangent line as slope of curve
:width: 300px
:align: center
```

How can we estimate the slope of the line at $x_0$? First, evaluate $f(x_0)$.
Then, evaluate $f(x_1)$ at another point nearby. The slope is:

\begin{equation}
\frac{f(x_1)-f(x_0)}{x_1-x_0}
\end{equation}

If we make $x_1$ very close to $x_0$, we will get the slope of the tangent line.
This suggests uses of a limit! We call the slope of the tangent to *f* at $x$
the *derivative* of *f*.

```{topic} Derivative as a limit
The first derivative $f'(x)$ of a function $f(x)$ is:

\begin{equation}
f'(x)= \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}
\end{equation}

The derivative is also commonly notated as $\d{f}/\d{x}$ or $\dot f$.
```

The derivative represents how fast the function is changing (rate of change).
This is especially useful in physics when the function represents a coordinate,
and the independent variable is time.

````{example} Ball moving with constant velocity
A ball has position coordinate *x* and is moving with constant velocity *v*, so

\begin{equation}
x(t) = x_0 + vt
\end{equation}

where $x_0$ is the initial position that it starts from.

```{image} ./_images/ball.png
:alt: Ball with constant velocity example
:width: 300px
:align: center
```

Show that the first derivative of the position *x* is the velocity *v*.

---

\begin{align}
x'(t) &= \lim_{h\to 0} \frac{x(t+h)-x(t)}{h} \\
&= \lim_{h\to 0} \frac{[x_0+v(t+h)]-[x_0+vt]}{h} \\
&=\lim_{h\to 0} \frac{vh}{h} \\
&= v
\end{align}
````

## Formal definition

The derivative is also a function!

```{image} ./_images/derivative_function.png
:alt: Function and derivative
:align: center
:width: 300px
```

For example, if $f(x) = x^2$,
\begin{align}
f'(x) &= \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \\
    &= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\
    &= \lim_{h \to 0} \frac{2xh + h^2}{h} \\
    &= \lim_{h \to 0} (2x + h) \\
    &= 2x
\end{align}

This function represents how the slope of the line tangent to *f* is changing
with *x*. For this function, the slope continually increases! Some additional
examples:

1. $f(x) = (x - 1)^2 + 1$

   ```{solution}
   \begin{align}
   f'(x) &= \lim_{h \to 0} \frac{[(x + h - 1)^2 + 1] - [(x - 1)^2 + 1]}{h} \\
       &= \lim_{h \to 0} \frac{(x - 1)^2 +
          2(x - 1)h + h^2 + 1 - (x - 1)^2 - 1}{h} \\
       &= \lim_{h \to 0} \frac{2(x - 1)h + h^2}{h} \\
       &= \lim_{h \to 0} 2(x - 1) + h \\
       &= 2(x - 1)
   \end{align}
   ```

2. $f(x) = 1/x$

   ```{solution}
   \begin{align}
   f'(x) &= \lim_{h \to 0} \frac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h} \\
       &= \lim_{h \to 0} \frac{\dfrac{x - (x + h)}{x(x+h)}}{h}  \\
       &= \lim_{h \to 0} \frac{\dfrac{- h}{x(x+h)}}{h}  \\
       &= \lim_{h \to 0} \frac{-1}{(x+h)x} \\
       &= \frac{-1}{x^2}
   \end{align}
   ```

## Differentiability

To be differentiable, this limit must exist. A function would not be
differentiable at a jump, cusp, coner, or vertical tangent

```{image} ./_images/nondifferentiable.jpg
:alt: Function and derivative
:align: center
:width: 400px
```

## Higher-order derivatives

We can try to take the derivative of any function, including the derivative
itself! For example, consider $f(x) = x^2$, which has $f'(x) = 2x$. We can
evaluate the derivative of $f'$, which we will denote $f''$ using the same
formal definition:

\begin{align}
f''(x) &= \lim_{h \to 0}{\frac{f'(x+h)-f'(x)}{h}} \\
    &= \lim_{h \to 0}{\frac{2(x+h)-2x}{h}} \\
    &= \lim_{h \to 0}{\frac{2h}{2}} \\
    &= 2
\end{align}

We call $f''$ the second derivative of *f*. Repeating the process we can obtain
even higher-order derivatives. These may be denoted by additional primes
(e.g., $f'''$), but often by a superscript (e.g., $f^{(3)}$) since it gets
clunky to count too many primes.

Higher-order derivatives tell us about how lower-order derivatives are
changing. For example, the second derivative tells us how the first derivative
changes.

In a physical setting, we may have names for these different derivatives. For
example, if we have the position *x*, the velocity *v* is the first derivative
of *x*, while the acceleration *a* is the first derivative of *v* and second
derivative of *x*.

| Position | Velocity | Acceleration |
|----------|----------|--------------|
|    x     |    x'    |      x''     |
|          |    v     |      v'      |
|          |          |      a       |

````{example} Ballistic motion
Consider the two-dimensional ballistic motion:

```{image} ./_images/ballistic.jpg
:alt: Ballistic motion
:align: center
:width: 300px
```

The equations of motion for the horizontal position *x* and vertical position
*y* are:

\begin{align}
x(t) &= u_0 t &  \\
y(t) &= v_0 t - \frac{1}{2} g t^2
\end{align}

where $u_0$ and $v_0$ are the initial *x* and *y* components of the velocity,
and *g* is the acceleration due to gravity. Find the velocity and acceleration
in *x* and *y*.

---

In the *x* direction, call the velocity *u*. Then,

\begin{equation}
u = x' = u_0
\end{equation}

Since *u* is a constant, the first derivative of *u* (second derivative of *x*)
must be zero, and there is no acceleration in the *x* direction.

In the *y* direction, call the velocity *v*. Then,

\begin{align}
v &= y' = v_0 - gt \\
a &= v' = y'' = -g
\end{align}

The velocity in *y* is constantly increasing from the constant acceleration *a*
due to gravity. This makes sense from what we know from physics!
````