MTH 251 – Differential Calculus Chapter 3 – DifferentiationDerivative of a ConstantDerivative of a Positive Integer Power of xDerivative of a Constant Multiple of f(x)Derivative of a SumDerivative of a DifferenceDerivative of a PolynomialDerivative of an ExponentialDerivative of the Exponential FunctionDerivative of a ProductSlide 11Derivative of a QuotientSlide 13Derivative of a Negative Integer PowerHigher-Order DerivativesHigher-Order Derivatives of PolynomialsMTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.2Differentiation Rules forPolynomials, Exponentials,Products, and QuotientsDerivative of a Constant•Proof …[ ]0dcdx=0limhc ch�-=0=0lim 0h�=0( ) ( )limhf x h f xh�+ -Derivative of a Positive Integer Power of x•Proof …1n ndx nxdx-� �=� �1 20( )limnhnx h hh-�+=ggg1nnx-=0( ) ( )limhf x h f xh�+ -0( )limn nhx h xh�+ -=10lim ( )nhnx h-�� �= +� �gggn is a positive integerDerivative of a Constant Multiple of f(x)•Proof …[ ] [ ]( ) ( )d dcf x c f xdx dx=[ ]( )dc f xdx=0( ) ( )limhcf x h cf xh�+ -0( ) ( )limhf x h f xch�+ -=Derivative of a Sum•Proof …[ ] [ ] [ ]( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx+ = +[ ] [ ]( ) ( )d df x g xdx dx= +[ ] [ ]0( ) ( ) ( ) ( )limhf x h g x h f x g xh�+ + + - +0( ) ( ) ( ) ( )limhf x h f x g x h g xh h�+ - + -� �= +� �� �0 0( ) ( ) ( ) ( )lim limh hf x h f x g x h g xh h� �+ - + -= +Derivative of a Difference•Proof …[ ] [ ] [ ]( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx- = -[ ] [ ]( ) ( )d df x g xdx dx= -[ ] [ ]0( ) ( ) ( ) ( )limhf x h g x h f x g xh�+ - + - -0( ) ( ) ( ) ( )limhf x h f x g x h g xh h�+ - + -� �= -� �� �0 0( ) ( ) ( ) ( )lim limh hf x h f x g x h g xh h� �+ - + -= -Derivative of a Polynomial•Just apply all of the previous rules …•Example …5 32 4 7dx x xdx� �- + -� �Derivative of an Exponential•Proof …0( ) ( )limhf x h f xh�+ -lnx xda a adx� �=� �0limx h xha ah+�-=0limx h xha a ah�-=01limhxhaah�-=lnxa a=This last step will be investigated in chapter 7.Derivative of the Exponential Function•Proof … use the last formula with a = eRemember: ln e = 1x xde edx� �=� �Derivative of a Product•Proof …0( ) ( ) ( ) ( )limhf x h g x h f x g xh�+ + -[ ]( ) ( ) '( ) ( ) ( ) '( )df x g x f x g x f x g xdx= +0( ) (( ) ( ) ( ) ( )lim) ( ) ( )hf x h g x f x h g xf x h g x h f x g xh�+ + -=- + + +[ ] [ ]0( ) ( ) ( ) ( ) ( ) ( )limhf x h g x h g x f x h f x g xh�+ + - + + -=[ ] [ ]0 0( ) ( ) ( ) ( ) ( ) ( )lim limh hf x h g x h g x f x h f x g xh h� �+ + - + -= +( ) '( ) '( ) ( )f x g x f x g x= +Derivative of a Product•Example …[ ]( ) ( ) '( ) ( ) ( ) '( )df x g x f x g x f x g xdx= +2 xdx edx� �� �Derivative of a Quotient•Proof … almost the same as the product rule.•Memory device …“lo-de-hi; hi-de-lo; square the bottom and away we go”[ ]2( ) '( ) ( ) ( ) '( )( )( )d f x f x g x f x g xdx g xg x� �-=� �� �[ ] [ ][ ]2( ) ( ) ( ) ( )( )( )( )d dg x f x f x g xd f xdx dxdx g xg x-� �=� �� �[ ] [ ][ ]2( ) ( ) ( ) ( )( )( )( )x xxg x D f x f x D g xf xDg xg x-� �=� �� �Derivative of a Quotient•Example …245d xdx x� �-� �� �[ ]2( ) '( ) ( ) ( ) '( )( )( )d f x f x g x f x g xdx g xg x� �-=� �� �Derivative of a Negative Integer Power•Proof … let m = –n (i.e. m is a positive integer) 1n ndx nxdx-� �=� �1nmd dxdx dx x� �� �=� �� �� �n is a negative integer[ ]21 1m mmd dx xdx dxx� �-� �=� �� �[ ]120 1m mmx mxx-� �-� �=12mmmxx-� �-� �=1mmx- -=-1nnx-=Higher-Order Derivatives•The derivative of the derivative is called the second derivative.•The derivative of the derivative of the derivative is called the third derivative.•Etc …[ ]2 22 2( ) ''( ) ''d d yf x f x ydx dx= = =[ ]3 33 3( ) '''( ) '''d d yf x f x ydx dx= = =[ ]( ) ( )( ) ( )n nn nn nd d yf x f x ydx dx= = =Higher-Order Derivatives of Polynomials•Example … find y’, y’’, y’’’, y(4), y(5) for …•y’ =•y’’ =•y’’’ =•y(4) =•y(5) =•General rule for nth degree polynomials.nth derivative is an! (a is the leading coefficient)All derivatives higher than the nth order are zero.3 23 2 1y x x x= + -