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Determine the derivatives of the following at any point by using first principles.

a. f(x) = 2x

b. f(x) = 3 - x²

c. f(x) = 2/x

d. g(x) = -3/x

Answer :

To find the derivative of a function using first principles, we use the definition of the derivative:

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

Let's apply this to each function given:

a. [tex]f(x) = 2x[/tex]


  1. Compute [tex]f(x+h)[/tex]:
    [tex]f(x+h) = 2(x+h) = 2x + 2h[/tex]


  2. Substitute into the derivative formula:
    [tex]f'(x) = \lim_{{h \to 0}} \frac{2x + 2h - 2x}{h} = \lim_{{h \to 0}} \frac{2h}{h} = \lim_{{h \to 0}} 2 = 2[/tex]



So, [tex]f'(x) = 2[/tex].

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


  1. Compute [tex]f(x+h)[/tex]:
    [tex]f(x+h) = 3 - (x+h)^2 = 3 - (x^2 + 2xh + h^2) = 3 - x^2 - 2xh - h^2[/tex]


  2. Substitute into the derivative formula:
    [tex]f'(x) = \lim_{{h \to 0}} \frac{3 - x^2 - 2xh - h^2 - (3 - x^2)}{h} = \lim_{{h \to 0}} \frac{-2xh - h^2}{h}[/tex]
    [tex]= \lim_{{h \to 0}} (-2x - h) = -2x[/tex]



So, [tex]f'(x) = -2x[/tex].

c. [tex]f(x) = \frac{2}{x}[/tex]


  1. Compute [tex]f(x+h)[/tex]:
    [tex]f(x+h) = \frac{2}{x+h}[/tex]


  2. Substitute into the derivative formula:
    [tex]f'(x) = \lim_{{h \to 0}} \frac{\frac{2}{x+h} - \frac{2}{x}}{h} = \lim_{{h \to 0}} \frac{2x - 2(x+h)}{h(x+h)x}[/tex]
    [tex]= \lim_{{h \to 0}} \frac{-2h}{h(x^2 + xh)} = \lim_{{h \to 0}} \frac{-2}{x^2 + xh} = -\frac{2}{x^2}[/tex]



So, [tex]f'(x) = -\frac{2}{x^2}[/tex].

d. [tex]g(x) = -\frac{3}{x}[/tex]


  1. Compute [tex]g(x+h)[/tex]:
    [tex]g(x+h) = -\frac{3}{x+h}[/tex]


  2. Substitute into the derivative formula:
    [tex]g'(x) = \lim_{{h \to 0}} \frac{-\frac{3}{x+h} + \frac{3}{x}}{h} = \lim_{{h \to 0}} \frac{-3x + 3(x+h)}{h(x+h)x}[/tex]
    [tex]= \lim_{{h \to 0}} \frac{3h}{h(x+h)x} = \lim_{{h \to 0}} \frac{3}{-x^2 - xh} = \frac{3}{x^2}[/tex]



So, [tex]g'(x) = \frac{3}{x^2}[/tex].

These calculations show how the derivative is found using first principles, breaking down each function to find its rate of change at any point.

Thanks for taking the time to read Determine the derivatives of the following at any point by using first principles a f x 2xb f x 3 x²c f x 2 xd. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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