2.15: Homework- Examples of the Definition of the Derivative

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Simplify each expression involving fractions or rational expressions.
1. $(x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)$
  $=\frac{x+1}{3} + x = \frac{4x+1}{3}$
  ans
2. $\cfrac{\cfrac{1}{3} + 1}{1-\cfrac{1}{3}}$
  $2$
  ans
3. $\cfrac{x + 1}{\cfrac{1}{x}}$
  $x^2 + x$
  ans
4. $\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}$
  $\frac{1}{2x+1}$
  ans
5. $\cfrac{\cfrac{2}{x} - \cfrac{1}{x}}{\cfrac{1 - y}{y}}$
  $\frac{y}{x - xy}$
  ans
In each case, use the definition of the derivative to find $f'(x)$ (in other words, take the derivative!)
1. $f(x) = 3x - 5 $
  $3$
  ans
2. $f(x) = \frac{1}{2} x + 1 $
  $quicklatex.com-92759ad21f5aa105eb2ef212fdb68ffa_l3.png$
  ans
3. $f(x) = 2 x^2$
  $4x$
  ans
4. $f(x) = (x^2 + x)$
  $2x + 1$
  ans
5. $\frac{d}{dx} (e^x)$ (hint: from yesterday’s homework, we have $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$)
  $e^x$
  ans
6. $f(x) = x^3$
  $x^3$
  ans
In each case, use the definition of the derivative to find $f'(x)$ (in other words, take the derivative!). Each of these is like one of the “hard problems” (click here)
1. $f(x) = 2x^4$
  $8x^3$
  ans
2. $f(x) = \sqrt{2x}$
  $\frac{1}{\sqrt{2x}}$
  ans
3. $f(x) = \frac{2}{x} $
  $-\frac{2}{x^2}$
  ans
4. $f(x) = \sqrt{x+1}$
  $\frac{1}{2 \sqrt{x+1}}$
  ans
5. $f(x) = \frac{1}{x+1} $
  $-\frac{1}{(x+1)^2}$
  ans
6. $f(x) = \frac{1}{\sqrt{x}}$
  $\frac{-1}{2 x \sqrt{x}}$
  ans
Recall the derivative of $f(x) = x^2$ is given by $2x$.
1. Show that the derivative of $g(x) = x^2+1$ is $2x$ using the definition of the derivative. Can you find an intuitive reason why $f(x)$ and $g(x)$ would have the same derivative?
  Adding a constant moves the curve up or down, but that shift does not affect the slope of the tangent line
  ans
2. Find another function whose derivative is $2x$, other than $f(x)$ and $g(x)$.
  $x^2 + c$ for any value $c$
  ans