3.3: Basic Differentiation Formulas

It is tedious to evaluate \[f'(x) = \lim_{h\rightarrow 0} \dfrac{f(x + h) - f(x)}{h}\] every time. Let's look how we can reduce our work by looking at shortcuts.

Suppose $c, n \in \mathbb{R}$ and $f(x)$ and $g(x)$ are differentiable functions. Then

$\dfrac{d}{dx} (c) = 0$
$\dfrac{d}{dx} (x^n) = nx^{n-1} \ $ (called the Power Rule)
$\dfrac{d}{dx} (cf(x)) = c\dfrac{d}{dx}[f(x)]$
$\dfrac{d}{dx} [f(x) \pm g(x)] = \dfrac{d}{dx}[f(x)] \pm \dfrac{d}{dx}[g(x)]$

For each of the following functions $f(x)$, find $f'(x)$.

$f(x) = 28$
$f(x) = \pi^2$
$f(x) = x$
$f(x) = x^8$
$f(x) = x^2\sqrt{x}$
$f(x) = \sqrt{x}$
$f(x) = \dfrac{1}{\sqrt[3]{x}}$
$f(x) = 5x^3$
$f(x) = \dfrac{3}{\sqrt{x}}$
$f(x) = 4x^5 + 3x^4 - 8x^2 + x + 3$
$f(t) = \dfrac{t^2}{5} + \dfrac{5}{t^3}$

Exponential Functions

How about the derivative of $f(x) = e^x$?

\[\dfrac{d}{dx}(e^x) = e^x\]

Trigonometric Functions

How about the derivatives of trigonometric functions?

\[\dfrac{d}{dx} (\sin x) = \cos x \qquad \qquad \dfrac{d}{dx}(\cos x) = -\sin x\]

Find the derivative of $f(x) = 3e^x - \cos x + \sqrt[3]{x^2}$.

Find the derivative of $f(\theta) = \theta + \sin \theta$.

Higher-Order Derivatives

If $f(x)$ is a function, $f'(x)$ is also a function.

$f'(x)$ might have a derivative of it's own!

To find the second derivative, first find $f'(x)$, then find the derivative of $f'(x)$.

In general, this extends to the $n$th derivative. Here are common notations for multiple derivatives:

$f'(x), f''(x), f'''(x), \cdots, f^{(n)}(x)$
$D_xf(x), D^2_xf(x), D^3_xf(x), \cdots, D^{(n)}_xf(x)$
$y', y'', y''', \cdots, y^{(n)}$
$\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}, \dfrac{d^3y}{dx^3}, \cdots, \dfrac{d^ny}{dx^n}$

The last notation is called Leibniz notation.

Find the 27th derivative of $f(x) = \cos x$.

We can interpret the second derivative as the rate of change of a rate of change.

A familiar example of a second derivative is acceleration.

If you have a position function $s(t)$, then we saw $v(t) = s'(t)$ is the velocity.

The rate of change of velocity is acceleration, or in symbols: \[a(t) = v'(t) = s''(t)\]

An object hanging from a vertical spring is compressed upwards 4 centimeters from rest and released at $t = 0$. The position of the object at time $t$ is \[s = f(t) = 4\cos t\] Find the velocity and acceleration at time $t$.