Section 2.3, Differentiation Formulas

Differentiation Formulas

For each of these differentiation formulas, we’ll give it using Leibniz notation. We’ll often use the Newton/prime notation, and if at any time you have a question, please ask!

Formulas

1. Constant function: $\dfrac{d}{dx} (c) = 0$ if $c$ is a constant. The slope of the line $y=c$ is 0.
2. Basic linear function: $\dfrac{d}{dx} (x) = 1$ The slope of the line $y=x$ is 1.
3. The Power Rule $\dfrac{d}{dx} (x^n) = nx^{n-1}$ if $n$ is a non-zero integer.

Examples

• Find the derivative of $y = x^3$
  
  Click to reveal the answer.

  $\dfrac{dy}{dx} = 3 x^{2}
• Find the derivative of $y = x^{1551}$
  Click to reveal the answer.

  $\dfrac{dy}{dx} = 1551 x^{1550}$
• Find the derivative of $f(x) = \sqrt{x}$
  Click to reveal the answer.

  $f'(x) = \dfrac{1}{2} x^{-1/2} = \dfrac{1}{2x^{1/2}} = \dfrac{1}{2\sqrt{x}} $
• Find the derivative of $f(x) = \frac{1}{x^5} = x^{-5}$
  Click to reveal the answer.

  $f'(x)= -5 x^{-5-1} = \dfrac{-5}{x^{6}}$

Formulas, continued

4. Constant multiple rule: if $c$ is a constant, $$ \dfrac{d}{dx} (c f(x) ) = c \dfrac{d}{dx} f(x) $$

5. Sum Rule $$ \dfrac{d}{dx} (f(x) + g(x) ) = \dfrac{d}{dx} f(x) + \dfrac{d}{dx} g(x) $$

6. Difference Rule $$ \dfrac{d}{dx} (f(x) - g(x) ) = \dfrac{d}{dx} f(x) - \dfrac{d}{dx} g(x) $$

Examples

• Find the derivative of $f(x) = x^3 - x + 1$
  Click to reveal the answer.

  $f'(x) = 3x^2 - 1 + 0 = 3x^2 - 1$, just like we found in a previous section!

Formulas, continued

7. $\dfrac{d}{dx} (\sin x) = \cos x$

8. $\dfrac{d}{dx} (\cos x) = - \sin x$

Examples:

• Find the derivative of $g(x) = 2\sin x - 3\cos x$

  Click to reveal the answer.

  $g'(x) = 2\cos x - 3\cdot (-\sin x) = 2\cos x + 3 \sin x$

• If $f(x) = \sin x$, find $f’(x)$, $f’’(x)$, $f’’’(x)$, and $f^{(4)}(x)$

  Click to reveal the answer.

  $\begin{align*} f(x) &= \sin x \\ f'(x) &= \cos x \\ f''(x) &= - \sin x \\ f^{(4)}(x) &= -\cos x \\ f^{(5)}(x) &= - (-\sin x) = \sin x & \text{ not that I asked...}\end{align*}$ What pattern do you notice? What do you think $f^{(6)}(x)$ is? What about $f^{(25)}(x)$?

Formulas, continued,

I’m going to switch to Newtonian/prime notation for these because it’s much cleaner and clearer to write and read.

9. The Product Rule. $$ (fg)’ = f’g + fg' $$

   EXAMPLE Find the derivative of $f(x) = x\sin x$

10. The Quotient Rule. $$ \left[ \dfrac{f}{g} \right]’ = \dfrac{gf’ - fg’}{g^2} $$ EXAMPLE Find the derivative of $h(x) = \dfrac{x^2}{\cos x}$

Examples

• Compute the derivative of $f(x) = \tan x$

• Compute the derivative of $f(x) = \dfrac{1}{x^5}$ using quotient rule.

• Compute the derivative of $f(x) = \dfrac{1}{x^5}$ using the product rule.

Now practice!

Head over to WebAssign and work on section 2.3. If you have questions, let me know!