1Chapter 6Differential CalculusThe two basic forms of calculus aredifferential calculus and integral calculus.This chapter will be devoted to the formerand Chapter 7 will be devoted to thelatter. Finally, Chapter 8 will be devotedto a study of how MATLAB can be used forcalculus operations.
2Differentiation and the DerivativeThe study of calculus usually begins withthe basic definition of a derivative. Aderivative is obtained through the processof differentiation, and the study of all formsof differentiation is collectively referred toas differential calculus.If we begin with afunction and determine its derivative, wearrive at a new function called the firstderivative. If we differentiate the firstderivative, we arrive at a new functioncalled the second derivative, and so on.
3The derivative of a function is theslope at a given point.
4Various Symbols for the Derivative( )or ( ) ordy df xf xdx dx0Definition: limxdy ydx x∆ →∆=∆
11Development of a Simple Derivative2y x=2( )y y x x+ ∆ = + ∆2 22 ( )y y x x x x+ ∆ = + ∆ + ∆
12Development of a Simple DerivativeContinuation22 ( )y x x x∆ = ∆ + ∆2yx xx∆= + ∆∆0lim 2xdy yxdx x∆ →∆= =∆
13Chain Rule( )y f u= ( )u u x=( )( )dy df u du duf udx du dx dx= =( )( )df uf udu=where
14Example 6-2. Approximate the derivativeof y=x2at x=1 by forming small changes.2(1) (1) 1y = =2(1.01) (1.01) 1.0201y = =1.0201 1 0.0201y∆ = − =0.02012.010.01dy ydx x∆≈ = =∆
15Example 6-3. The derivative of sin uwith respect to u is given below.( )sin cosdu udu=Use the chain rule to find thederivative with respect to x of24siny x=
16Example 6-3. Continuation.2u x=2duxdx=2( )4(cos )(2 ) 8 cosdy du dy duf udx dx du dxu x x x= == =
17Table 6-1. Derivatives( )f x ( )f x Derivative Number( )af x ( )af x D-1( ) ( )u x v x+ ( ) ( )u x v x+ D-2( )f u ( )( )du df u duf udx du dx=D-3a 0 D-4( 0)nx n ≠ 1nnx − D-5( 0)nu n ≠ 1n dunudx−D-6uv dv duu vdx dx+ D-7uv2du dvv udx dxv− D-8ue u duedxD-9
23Higher-Order Derivatives( )y f x=( )( )dy df xf xdx dx= =2 22 2( )( )d y d f x d dyf xdx dx dx dx = = = ÷ 3 3 2(3)3 3 2( )( )d y d f x d d yf xdx dx dx dx = = = ÷
24Example 6-7. Determine the 2ndderivative with respect to x of thefunction below.5sin 4y x=5(cos4 ) (4 ) 20cos4dy dx x xdx dx= × =( )2220 sin 4 (4 ) 80sin 4d y dx x xdx dx= − × = −
25Applications: Maxima and Minima1. Determine the derivative.2. Set the derivative to 0 and solve forvalues that satisfy the equation.3. Determine the second derivative.(a) If second derivative > 0, point is aminimum.(b) If second derivative < 0, point is amaximum.
26DisplacementVelocityAccelerationdyvdt=Displacement, Velocity, and Accelerationy22dv d yadt dt= =
27Example 6-8. Determine local maximaor minima of function below.3 2( ) 6 9 2y f x x x x= = − + +23 12 9dyx xdx= − +23 12 9 0x x− + =1 and 3x x= =
28Example 6-8. Continuation.23 12 9dyx xdx= − +For x = 1, f”(1) = -6. Point is a maximum andymax= 6.For x = 3, f”(3) = 6. Point is a minimum andymin = 2.226 12d yxdx= −