Calculus II MATH 1920
pellissippi state community college
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Date Created: 11/01/15
THE DERIVATIVE Review Notes on Differentiation for MATH 1920 with Practice Exercises The Leibnitz notation for the derivative of fX with respect to X is dlax It is commonly used x whenever we are asked to nd the derivative but do not have the function given as an equation It is essential whenever it is not clear which letter in a function is the independent variable It was used to introduce every differentiation rule in MATH 1910 See pages 1926198202205 223224228 23424024124324424751268270 etc Several different notations forthe rst derivative are shown on p 162 Higher order derivatives and their notations are discussed on p 1658 Notation is very important because poor notation misleads others who try to follow your work and it frequently misleads you as well Do not put equal signs between quantities which are not equal The symbol d1 has no meaning because we do not know what to differentiate x Find and simplify each ofthe following derivatives 1 axzy3 34130 71 HINT The variable is a treat all other letters as constants d2 d d 2 23 4 3 234 4 234 dbz axy abc7r dyaxy abc7r dX axy abc7r Some derivatives can be found using different methods Whenever you are tempted to use a cumbersome method such as the product or quotient rule study the problem to see if an easier approach will work See note at bottom of p 205 No matter which method you use you must simplify the answer Find and simplify each ofthe following derivatives using the speci ed methods Sec 31 2 i w Use the quotient rule first Then divide the denominator into the dX J numerator and rework the problem using the power rule Showthat your answers are equivalent 6 3t 2 Use the product rule rst Then distribute the monomial and reworkthe problem using the power rule Show that your answers are equivalent Many differentiation problems require the use ofthe chain rule The chain rule is never used alone but is always used in combination with at least one other rule Some situations requiring the chain rule are quantities raised to powers roots ofquantities trig functions ofquantities exponents which are quantities etc Sometimes other methods can be employed butthe chain rule is usually easier and shorter The answer needs to be in factored form Find and simplify each ofthe following using the speci ed methods on 7 Sec 35 7 d1 3x 72 Expand the quantity and use the power rule Then rework the x problem using the chain rule d d y31 8 3X 2105X2 X112 9 dX dyy31 Trig functions and log functions must have arguments The words sin tan log have meaning in the English language but no mathematical meaning The tendency in these problems is to use the product rule where it does not apply Find and simplify each ofthe following derivatives Sec 31 34 35 37 d d d 3 2 10 costanx 11 smcostan2x 12 tan ax dX dX dX d xcosx d 2 x d 2 13 14 15 I 4 dX e dX X e dX 093X 16 15 17 linkeX dX dX It is essential that you learn to apply formulas for both derivatives and integrals because most problems are too complicated to do easily by hand While you do not have to memorize the rules for differentiating the inverse trig functions you must be able to apply them Find and simplify each ofthe following derivatives Sec 36 18 i tan 1e4quot 19 i sin 1x2 2 dX dr 20 Hyperbolic functions are discussed on p 2534 Use the de nitions given there to show that d1 sinhx cosh x This is very similarto which othertrig derivative x Limits are very important in calculus because it is from limits that we get derivatives While numerical tables and graphs may be helpful in some instances they can be very misleading in others Therefore they should not be used as the sole means of nding a limit There are several classic techniquestaught in Algebra II which are commonly used in nding limits These techniques include factoring rationalizing the numerator and simplifying compound fractions by multiplying both numerator and denominator by the LCD ofthe small fractions in the numerator of the compound fraction Rememberthatg is an indeterminate form and does not equal anything However ifyou get when you try to use direct substitution to nd a limit it meansthere probably is a limit It is your
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