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# lecture21hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 21) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 15

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Outline 1 General Power Rule 2 Other General Rules 3 Chain Rule Albert Ku (HKUST) MATH 006 2 / 15
General Power Rule General Power Rule Example Find the derivative of f ( x ) = (2 x + 1) 100 . Remark Theoretically, we can expand (2 x + 1) 100 into a polynomial and differentiate it term by term. But it is not practical because the power is too large. To differentiate this function efficiently, we need the general power rule , which provides us with the formula for differentiating a function that is a a power of another function. Albert Ku (HKUST) MATH 006 3 / 15

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General Power Rule Theorem (General Power Rule) If u ( x ) is a differentiable function and n is any real number, and f ( x ) = [ u ( x )] n . Then, f ( x ) = n [ u ( x )] n - 1 u ( x ) . Remarks Roughly speaking, we differentiate the function like the standard power rule (([ · ] n ) = n [ · ] n - 1 ) and then multiply it by the derviative of the expression inside the bracket. If u ( x ) = x , then u ( x ) = 1 and the formula becomes the standard power rule.
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