17 Homework: Examples of the Definition of the Derivative

  1. Simplify each expression involving fractions or rational expressions.
    1. [latex](x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)[/latex]
      [latex]=\frac{x+1}{3} + x = \frac{4x+1}{3}[/latex]

      ans
    2. [latex]\cfrac{\cfrac{1}{3} + 1}{1-\cfrac{1}{3}}[/latex]
      [latex]2[/latex]

      ans
    3. [latex]\cfrac{x + 1}{\cfrac{1}{x}}[/latex]
      [latex]x^2 + x[/latex]

      ans
    4. [latex]\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}[/latex]
      [latex]\frac{1}{2x+1}[/latex]

      ans
    5. [latex]\cfrac{\cfrac{2}{x} - \cfrac{1}{x}}{\cfrac{1 - y}{y}}[/latex]
      [latex]\frac{y}{x - xy}[/latex]

      ans
  2. In each case, use the definition of the derivative to find [latex]f'(x)[/latex] (in other words, take the derivative!)
    1. [latex]f(x) = 3x - 5[/latex]
      [latex]3[/latex]

      ans
    2. [latex]f(x) = \frac{1}{2} x + 1[/latex]
      [latex]1/2[/latex]

      ans
    3. [latex]f(x) = 2 x^2[/latex]
      [latex]4x[/latex]

      ans
    4. [latex]f(x) = (x^2 + x)[/latex]
      [latex]2x + 1[/latex]

      ans
    5. [latex]\frac{d}{dx} (e^x)[/latex] (hint: from yesterday’s homework, we have [latex]\lim_{h \to 0} \frac{e^h - 1}{h} = 1[/latex])
      [latex]e^x[/latex]

      ans
    6. [latex]f(x) = x^3[/latex]
      [latex]x^3[/latex]

      ans
  3. In each case, use the definition of the derivative to find [latex]f'(x)[/latex] (in other words, take the derivative!). Each of these is like one of the “hard problems” (click here)
    1. [latex]f(x) = 2x^4[/latex]
      [latex]8x^3[/latex]

      ans
    2. [latex]f(x) = \sqrt{2x}[/latex]
      [latex]\frac{1}{\sqrt{2x}}[/latex]

      ans
    3. [latex]f(x) = \frac{2}{x}[/latex]
      [latex]-\frac{2}{x^2}[/latex]

      ans
    4. [latex]f(x) = \sqrt{x+1}[/latex]
      [latex]\frac{1}{2 \sqrt{x+1}}[/latex]

      ans
    5. [latex]f(x) = \frac{1}{x+1}[/latex]
      [latex]-\frac{1}{(x+1)^2}[/latex]

      ans
    6. [latex]f(x) = \frac{1}{\sqrt{x}}[/latex]
      [latex]\frac{-1}{2 x \sqrt{x}}[/latex]

      ans
  4. Recall the derivative of [latex]f(x) = x^2[/latex] is given by [latex]2x[/latex].
    1. Show that the derivative of [latex]g(x) = x^2+1[/latex] is [latex]2x[/latex] using the definition of the derivative. Can you find an intuitive reason why [latex]f(x)[/latex] and [latex]g(x)[/latex] 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 [latex]2x[/latex], other than [latex]f(x)[/latex] and [latex]g(x)[/latex].
      [latex]x^2 + c[/latex] for any value [latex]c[/latex]

      ans

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