mat21b useful handouts.pdf - Here are some derivatives you...

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Here are some derivatives you really need to be aware of for calc 2. Note: a > 0 and a 6 = 1 and arcsin( x ) = sin - 1 ( x ) d dx ln x = 1 x d dx e x = e x d dx log a x = 1 x ln a d dx a x = a x ln a d dx ln[ f ( x )] = 1 f ( x ) · f 0 ( x ) = f 0 ( x ) f ( x ) d dx e [ f ( x )] = e [ f ( x )] · f 0 ( x ) d dx log a [ f ( x )] = 1 f ( x ) ln a · f 0 ( x ) = f 0 ( x ) f ( x ) ln a d dx a [ f ( x )] = a [ f ( x )] · f 0 ( x ) ln a d dx sin - 1 ( x ) = 1 1 - x 2 d dx cos - 1 ( x ) = - 1 1 - x 2 d dx tan - 1 ( x ) = 1 1 + x 2 d dx cot - 1 ( x ) = - 1 1 + x 2 d dx sec - 1 ( x ) = 1 x x 2 - 1 d dx sin - 1 ( x ) = - 1 x x 2 - 1 I didn’t write them, but you should also be aware of the chain rule variants of the derivatives of each of the inverse trig functions so that, for example, you can find the derivative of sin - 1 ( x 5 ) or the derivative of cot - 1 ( x 3 · 2 x ) You should also be aware that, depending on how sec - 1 ( x ) is defined, it’s derivative can alternatively be written as d dx sec - 1 ( x ) = 1 | x | x 2 - 1 . A similar fact is true for csc - 1 ( x ). For the inverse trig derivates an interesting observation can be made. Take for example d dx sin - 1 ( x ) = 1 1 - x 2 . If we let x = sinθ , then the 1 - x 2 in the derivative becomes 1 - x 2 = 1 - sin 2 θ = cos 2 θ . That is, it simplifies nicely. The same is true for any of the other derivatives of the inverse trig functions, if we let x be the corresponding trig function of θ then things simplify nicely. For sec - 1 ( x ), if we let x = sec θ then x 2 - 1 = sec 2 θ - 1 = tan 2 θ . Try it for yourself with one of the other ones.
Co-21B Antiderivative Rules Differentiation Rules Antiderivative Rules 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑝𝑝 = 𝑝𝑝𝑑𝑑 𝑝𝑝−1 Power � 𝑑𝑑 𝑝𝑝 𝑑𝑑𝑑𝑑 = 𝑑𝑑 𝑝𝑝+1 𝑝𝑝 + 1 + 𝐶𝐶 , 𝑝𝑝 ≠ − 1 𝑑𝑑 𝑑𝑑𝑑𝑑 ln( 𝑑𝑑 ) = 1 𝑑𝑑 1 𝑑𝑑 𝑑𝑑𝑑𝑑 = ln| 𝑑𝑑 | + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑎𝑎 𝑥𝑥 = ln( 𝑎𝑎 ) 𝑎𝑎 𝑥𝑥 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑎𝑎 𝑘𝑘𝑥𝑥 = 𝑘𝑘 ln( 𝑎𝑎 ) 𝑎𝑎 𝑥𝑥 Exponential � 𝑎𝑎 𝑘𝑘𝑥𝑥 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 ln( 𝑎𝑎 ) 𝑎𝑎 𝑘𝑘𝑥𝑥 + 𝐶𝐶 , 𝑎𝑎 > 0, 𝑎𝑎 ≠ 1 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑒𝑒 𝑥𝑥 = 𝑒𝑒 𝑥𝑥 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑒𝑒 𝑘𝑘𝑥𝑥 = 𝑘𝑘𝑒𝑒 𝑘𝑘𝑥𝑥 � 𝑒𝑒 𝑘𝑘𝑥𝑥 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 𝑒𝑒 𝑘𝑘𝑥𝑥 + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 sin( 𝑑𝑑 ) = cos( 𝑑𝑑 ) 𝑑𝑑 𝑑𝑑𝑑𝑑 sin( 𝑘𝑘𝑑𝑑 ) = 𝑘𝑘 cos( 𝑘𝑘𝑑𝑑 ) Trig cos( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 sin( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 cos( 𝑑𝑑 ) = sin( 𝑑𝑑 ) 𝑑𝑑 𝑑𝑑𝑑𝑑 cos( 𝑘𝑘𝑑𝑑 ) = −𝑘𝑘 sin( 𝑘𝑘𝑑𝑑 ) sin( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 cos( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 tan( 𝑑𝑑 ) = sec 2 ( 𝑑𝑑 ) sec 2 ( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 tan( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 cot( 𝑑𝑑 ) = csc 2 ( 𝑑𝑑 ) csc 2 ( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 cot( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 sec( 𝑑𝑑 ) = sec( 𝑑𝑑 ) tan( 𝑑𝑑 ) sec( 𝑘𝑘𝑑𝑑 ) tan( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 sec( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 csc( 𝑑𝑑 ) = csc( 𝑑𝑑 ) cot( 𝑑𝑑 ) csc( 𝑘𝑘𝑑𝑑 ) cot( 𝑘𝑘𝑑𝑑 ) 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 csc( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 sin −1 ( 𝑑𝑑 ) = 1 1 − 𝑑𝑑 2 | 𝑑𝑑 | < 1 Inverse Trig 1 1 − 𝑘𝑘 2 𝑑𝑑 2 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 sin −1 ( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 tan −1 ( 𝑑𝑑 ) = 1 1 + 𝑑𝑑 2 1 1 + 𝑘𝑘 2 𝑑𝑑 2 𝑑𝑑𝑑𝑑 = 1 𝑘𝑘 tan −1 ( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 𝑑𝑑 𝑑𝑑𝑑𝑑 sec −1 ( 𝑑𝑑 ) = 1 | 𝑑𝑑 | √𝑑𝑑 2 1 | 𝑑𝑑 | > 1 1 𝑑𝑑√𝑘𝑘 2 𝑑𝑑 2 1 𝑑𝑑𝑑𝑑 = sec −1 ( 𝑘𝑘𝑑𝑑 ) + 𝐶𝐶 , 𝑘𝑘𝑑𝑑 > 1
Important Inverse Trig derivatives you need to know: (notice that sometimes I use “arc” and sometimes “to the -1,” they both mean the same thing) d dx arcsin( x ) = 1 1 - x 2 d dx arccos( x ) = - 1 1 - x 2 d dx arctan( x ) = 1 1+ x 2 d dx cot - 1 ( x ) = - 1 1+ x 2 d dx sec - 1 ( x ) = 1 | x | x 2 - 1 d dx csc - 1 ( x ) = - 1 | x | x 2 - 1 (notice that the left side and right side derivatives only differ by a negaitve sign) Also if we want to write them for when we use the chain rule (here u is a function of x ) we get: d dx arcsin( u ) = 1 1 - u 2 · du dx d dx arccos( u ) = - 1 1 - u 2 · du dx d dx arctan( u ) = 1 1+ u 2 · du dx d dx cot - 1 ( u ) = - 1 1+ u 2 · du dx d dx sec - 1 ( u ) = 1 | u | u 2 - 1 · du dx d dx csc - 1 ( u ) = - 1 | u | u 2 - 1 · du dx