lecture06

# lecture06 - 1 Formulas Remember mathematics is all about...

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1 Formulas Remember, mathematics is all about being lazy. Using the limit deﬁnition of the derivative will get very diﬃcult if we have to do it every time. Fortunately, we have methods for computing common derivatives like we have methods for computing common limits. d dx ( c ) Prove this. d dx ( x ) = 1 Prove this. d dx ( x n ) = nx n - 1 Prove this using the binomial theorem. 1.1 Derivative laws (like limit laws) d dx ( cf ( x )) d dx ( f ( x ) + g ( x )) d dx ( f ( x ) - g ( x )) Use scalar multiplication + addition. Example 1.1. Find f 0 ( x ) if f ( x ) = x 8 + 7 x 5 - 20 x 3 + 17 x 2 - 1 Example 1.2. Find the points of y = x 4 - 6 x 2 + 4 where the tangent line is horizontal. 1.2 Product and Quotient rules Product Rule Pull out and f ( x + h ) and a g ( x ). Quotient Rule Add and subtract f ( x ) g ( x ) Example 1.3. Find f 0 ( x ) if f ( x ) = (6 x 3 )(7 x 4 ) Example 1.4. Find d dx ± x 2 + x - 2 x 3 +6 ² 1.3 Power Functions Prove d dx ( x - n ) = - n - n - 1 using quotient rule. Example 1.5.

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lecture06 - 1 Formulas Remember mathematics is all about...

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