Math  /  Calculus

Questionf(x)=3x1f(x)=\sqrt{3 x-1}
Calculate the difference quotient: f(x+h)f(x)h\frac{f(x+h)-f(x)}{h}

Studdy Solution
Simplify the expression if possible. To simplify, we can multiply the numerator and the denominator by the conjugate of the numerator:
3x+3h13x1h3x+3h1+3x13x+3h1+3x1 \frac{\sqrt{3x + 3h - 1} - \sqrt{3x - 1}}{h} \cdot \frac{\sqrt{3x + 3h - 1} + \sqrt{3x - 1}}{\sqrt{3x + 3h - 1} + \sqrt{3x - 1}}
This simplifies the numerator using the difference of squares:
=(3x+3h1)(3x1)h(3x+3h1+3x1) = \frac{(3x + 3h - 1) - (3x - 1)}{h(\sqrt{3x + 3h - 1} + \sqrt{3x - 1})}
=3hh(3x+3h1+3x1) = \frac{3h}{h(\sqrt{3x + 3h - 1} + \sqrt{3x - 1})}
Cancel h h from the numerator and the denominator:
=33x+3h1+3x1 = \frac{3}{\sqrt{3x + 3h - 1} + \sqrt{3x - 1}}
The simplified difference quotient is:
33x+3h1+3x1 \frac{3}{\sqrt{3x + 3h - 1} + \sqrt{3x - 1}}

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