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
  1. $\dfrac{d}{dx} (c) = 0$
  2. $\dfrac{d}{dx} (x^n) = nx^{n-1} \ $ (called the Power Rule)
  3. $\dfrac{d}{dx} (cf(x)) = c\dfrac{d}{dx}[f(x)]$
  4. $\dfrac{d}{dx} [f(x) \pm g(x)] = \dfrac{d}{dx}[f(x)] \pm \dfrac{d}{dx}[g(x)]$
For each of the following functions $f(x)$, find $f'(x)$.
  1. $f(x) = 28$
  2. $f(x) = \pi^2$
  3. $f(x) = x$
  4. $f(x) = x^8$
  5. $f(x) = x^2\sqrt{x}$
  6. $f(x) = \sqrt{x}$
  7. $f(x) = \dfrac{1}{\sqrt[3]{x}}$
  8. $f(x) = 5x^3$
  9. $f(x) = \dfrac{3}{\sqrt{x}}$
  10. $f(x) = 4x^5 + 3x^4 - 8x^2 + x + 3$
  11. $f(t) = \dfrac{t^2}{5} + \dfrac{5}{t^3}$