High School

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A. Use first principles to determine the derivative of f if:

1. f(x) = x² + 3
2. f(x) = x²
3. f(x) = -2x²
4. f(x) = -3x
5. f(x) = 2/x

Answer :

To find the derivative of a function using first principles (also known as the limit definition of the derivative), we use the following formula:

[tex]f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]

We will apply this formula to each of the given functions:

  1. For [tex]f(x) = x^2 + 3[/tex]:

    • First, calculate [tex]f(x + h) = (x + h)^2 + 3 = x^2 + 2xh + h^2 + 3[/tex].

    • Substitute into the formula:

      [tex]f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2 + 3) - (x^2 + 3)}{h}[/tex]

    • Simplify:

      [tex]f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h)[/tex]

    • As [tex]h \to 0[/tex], the derivative [tex]f'(x) = 2x[/tex].

  2. For [tex]f(x) = x^2[/tex]:

    • Calculate [tex]f(x + h) = (x + h)^2 = x^2 + 2xh + h^2[/tex].

    • Substitute into the formula:

      [tex]f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h}[/tex]

    • Simplify:

      [tex]f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h)[/tex]

    • As [tex]h \to 0[/tex], the derivative [tex]f'(x) = 2x[/tex].

  3. For [tex]f(x) = -2x^2[/tex]:

    • Calculate [tex]f(x + h) = -2(x + h)^2 = -2(x^2 + 2xh + h^2) = -2x^2 - 4xh - 2h^2[/tex].

    • Substitute into the formula:

      [tex]f'(x) = \lim_{h \to 0} \frac{-2x^2 - 4xh - 2h^2 + 2x^2}{h}[/tex]

    • Simplify:

      [tex]f'(x) = \lim_{h \to 0} \frac{-4xh - 2h^2}{h} = \lim_{h \to 0} (-4x - 2h)[/tex]

    • As [tex]h \to 0[/tex], the derivative [tex]f'(x) = -4x[/tex].

  4. For [tex]f(x) = -3x[/tex]:

    • Calculate [tex]f(x + h) = -3(x + h) = -3x - 3h[/tex].

    • Substitute into the formula:

      [tex]f'(x) = \lim_{h \to 0} \frac{-3x - 3h + 3x}{h}[/tex]

    • Simplify:

      [tex]f'(x) = \lim_{h \to 0} \frac{-3h}{h} = \lim_{h \to 0} (-3)[/tex]

    • The derivative [tex]f'(x) = -3[/tex].

  5. For [tex]f(x) = \frac{2}{x}[/tex]:

    • Calculate [tex]f(x + h) = \frac{2}{x + h}[/tex].

    • Substitute into the formula:

      [tex]f'(x) = \lim_{h \to 0} \frac{\frac{2}{x + h} - \frac{2}{x}}{h}[/tex]

    • Combine the fractions:

      [tex]f'(x) = \lim_{h \to 0} \frac{2(x - (x + h))}{h(x)(x + h)} = \lim_{h \to 0} \frac{-2h}{hx(x + h)}[/tex]

    • Simplify:

      [tex]f'(x) = \lim_{h \to 0} \frac{-2}{x(x + h)}[/tex]

    • As [tex]h \to 0[/tex], the derivative [tex]f'(x) = \frac{-2}{x^2}[/tex].

These are the derivatives of the given functions. This process uses the concept of limits to find the rate of change of each function at any point [tex]x[/tex].

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