Find the difference quotient of f; that is, find [f(x+h)-f(x)] / h for the following function f(x)= 5 / x²

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justaguide | College Teacher | (Level 2) Distinguished Educator

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For the function `f(x) = 5/x^2` , the value `(f(x+h)-f(x))/h` is required.

 `(f(x+h)-f(x))/h`

= `(5/(x+h)^2 - 5/x^2)/h`

= `(5*x^2 - 5*(x+h)^2)/(h*(x+h)^2*x^2)`

The difference quotient is `(5*x^2 - 5*(x+h)^2)/(h*(x+h)^2*x^2)```

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mathformarines | Student, Undergraduate | eNotes Newbie

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f(x) = 5 / x²

f(x + h) = 5 / (x + h)²

f(x + h) - f(x) = [5 / (x + h)²] - [5 / x²]

= [5x² - 5(x + h)²] / [x²(x + h)²] ... taking a common denominator

= [5x² - 5(x² + 2hx + h²)] / [x²(x + h)²]

= [5x² - 5x² - 10hx - 5h²] / [x²(x + h)²]

= [-10hx - 5h²] / [x²(x + h)²]


so [f(x + h) - f(x)] / h = [-10hx - 5h²] / [x²(x + h)²] * 1/h

= [-10hx - 5h²] / [hx²(x + h)²]

= [-10x - 5h] / [x²(x + h)²]

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