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Power rule

The power rule for deriving a power function is

 

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Explanation

We use the function

what you can also write as

Differentiating with the product rule gives

So we can easily see that in general

 

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Example 1

For the function

f (x) = x³ + 4x² + x − 6

you can calculate the derivative as

f ′(x) = 3x² + 8x + 1

and at the point x = 3 this gives

f ′(3) = 3·3² + 8·3 + 1 = 27 + 24 + 1 = 52

 

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Example 2

For the function

f (x) = x³ + 4x² + x − 6

you can calculate the quadratic differential as

f ″(x) = 6x + 8

and at the point x = 3 this gives

f ″(3) = 6·3 + 8 = 18 + 8 = 26

 

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