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1、These icons indicate that teachers notes or useful web addresses are available in the Notes Page.,This icon indicates the slide contains activities created in Flash.These activities are not editable.,For more detailed instructions,see the Getting Started presentation.,Boardworks Ltd 2006,1 of 53,A-L
2、evel Maths:Core 3for OCR,C3.5 Differentiation and Integration 1,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,2
3、 of 53,The chain rule,Review of differentiation,So far,we have used differentiation to find the gradients of functions made up of a sum of multiples of powers of x.We found that:,and when xn is preceded by a constant multiplier k we have:,Review of differentiation,We will now look at how to differen
4、tiate exponential,logarithmic and trigonometric functions.,We will also look at techniques that can be used to differentiate:,Compound functions of the form f(g(x).For example:,Products of the form f(x)g(x),such as:,Quotients of the form,such as:,The chain rule,The chain rule is used to differentiat
5、e composite functions.,For instance,suppose we want to differentiate y=(2x+1)3 with respect to x.,One way to do this is to expand(2x+1)3 and differentiate it term by term.,Using the binomial theorem:,Differentiating with respect to x:,The chain rule,Another approach is to use the substitution u=2x+1
6、 so that we can write y=(2x+1)3 as y=u3.,The chain rule states that:,So if y=u3 whereu=2x+1,Using the chain rule:,The chain rule,Using the chain rule,Use the chain rule to differentiate y=with respect to x.,Let where u=3x2 5,The chain rule,Using the chain rule:,Let,Find given that.,where u=7 x3,The
7、chain rule using function notation,With practice some of the steps in the chain rule can be done mentally.,Suppose we have a composite function,y=g(f(x),If we lety=g(u)whereu=f(x),Using the chain rule:,But u=f(x)so,The chain rule,All of the composite functions we have looked at so far have been of t
8、he form y=(f(x)n.,In general,using the chain rule,Find the equation of the tangent to the curve y=(x4 3)3 at the point(1,8).,The chain rule,When x=1,Using y y1=m(x x1)the equation of the tangent at the point(1,8)is:,y+8=48(x 1),y=48x 48 8,y=48x 56,y=8(6x 7),y=(x4 3)3,The chain ruleConnected rates of
9、 changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,12 of 53,Connected rates of change,Rates of change,When we talk about rates of change we are usua
10、lly talking about the rate at which a variable changes with respect to time.,Suppose,for example,that a spherical balloon is slowly being inflated.,There are several changes we could measure,such as:,We can connect these rates of change using the chain rule.,The rate at which the radius r changes,Th
11、e rate at which the surface area A changes,The rate at which the volume V changes,For example,the rate of change of the surface area is connected to the rate of change of the radius by,Rates of change,So,the rate of change of the surface area is connected to the rate of change of the radius by,If th
12、e surface area of the balloon is increasing at a rate of 15 cm2 s1,find the rate at which the radius of the balloon is increasing at the moment when the radius is 4 cm.,We can find by differentiating the formula for the surfacearea of a sphere with respect to the radius.,A=4r2so,Connected rates of c
13、hange,We are given that=15 and we want to find when r=4.,Using we have:,=0.149,So,the radius is increasing at a rate of 0.149 cm s1(to 3 s.f.)at the moment when the radius of the balloon is 4 cm.,Connected rates of change,So,If the air in the fully-inflated balloon is released at a rate of 30 cm3 s1
14、,find the rate at which the surface area is decreasing at the moment when the radius is 5 cm.,Using the chain rule,the rate of change of the volume is connected to the rate of change of the radius by:,We are given that=30 and we want to find when r=5.,Connected rates of change,Here=30 and r=5:,We ca
15、n now find using with r=5:,4,=12,So,the surface area is decreasing at a rate of 12 cm2 s1 at the moment when the radius of the balloon is 5 cm.,Connected rates of change,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and re
16、lated functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,19 of 53,The relationship between,The relationship between,Suppose we are given x as a function of y instead of y as a function of x.For instance,x=4y2,We can find by differentiating with respect
17、 to y:,Using the chain rule we can write,So by the above result,if 8y then,from which we get:,The relationship between,Find the gradient of the curve with equation x=2y3 3y 7 at the point(3,2).,x=2y3 3y 7,At the point(3,2),y=2:,We can now find the gradient using the fact that,Differentiating inverse
18、 functions,Find,writing your answer in terms of x.,Let y=sin1 x so,The result is particularly useful for differentiatinginverse functions.For example:,x=sin y,Using the identity cos2y=1 sin2y,But sin y=x so,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related
19、 functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,23 of 53,Differentiating ex and related functions,The derivative of ex,From this,it follows that,A special property of the exponential function ex is that,whe
20、re k is a constant.,For example,if y=4ex x3,Functions of the form ekx,Suppose we are asked to differentiate a function of the form ekx,where k is a constant.For example,Differentiate y=e5x with respect to x.,Using the chain rule:,Let where u=5x,In practice,we wouldnt need to include this much workin
21、g.,Functions of the form ekx,We would just remember that in general,For example,We can use the chain rule to extend this to any function of the form ef(x).,Functions of the form ef(x),If y=ef(x)then we can let,Using the chain rule:,Let where u=f(x),then,So in general,In words,to differentiate an exp
22、ression of the form y=ef(x)we multiply it by the derivative of the exponent.,Functions of the form ef(x),For example,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExami
23、nation-style questions,Contents,Boardworks Ltd 2006,29 of 53,Differentiating ln x and related functions,The derivative of ln x,Remember,ln x is the inverse of ex.,So,if y=ln x,then x=ey,Differentiating with respect to y gives:,But ey=x so,Functions of the form ln kx,Suppose we want to differentiate
24、a function of the form ln kx,where k is a constant.For example:,Differentiate y=ln 3x with respect to x.,Using the chain rule:,Let where u=3x,Functions of the form ln kx,When functions of the form ln kx are differentiated,the ks will always cancel out,so in general,We can use the chain rule to exten
25、d to functions of the more general form y=ln f(x).,Using the chain rule:,Let where u=f(x),then,Functions of the form ln(f(x),In general,using the chain rule,For example,Functions of the form ln(f(x),In some cases we can use the laws of logarithms to simplify a logarithmic function before differentia
26、ting it.,Remember that,ln(ab)=ln a+ln b,ln an=n ln a,Differentiate with respect to x.,Functions of the form ln(f(x),ln 2 is a constant and so it disappears when we differentiate.,If we had tried to differentiate without simplifying it first,we would have had:,The derivative is the same,but the algeb
27、ra is more difficult.,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,36 of 53,The product rule,The product rule,
28、The product rule allows us to differentiate the product of two functions.,It states that if y=uv,where u and v are functions of x,then,So,Letu=x4 andv,The product rule,Using the product rule:,The product rule,Give the coordinates of any stationary points on the curve y=x2e2x.,So,Letu=x2 andv,Using t
29、he product rule:,The product rule,When x=0,y=(0)2e0,=0,The point(0,0)is a stationary point on the curve y=x2e2x.,When x=1,y=(1)2e2,=e2,The point(1,e2)is also a stationary point on the curve y=x2e2x.,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functio
30、nsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination-style questions,Contents,Boardworks Ltd 2006,41 of 53,The quotient rule,The quotient rule,The quotient rule allows us to differentiate the quotient of two functions.,It states that if y=,where u and v are functi
31、ons of x,then,Find given that.,Letu=2x+1andv=5x2,So,The quotient rule,The quotient rule,Letu=ln x4 andv=x2,So,Using the quotient rule:,Find the equation of the tangent to thecurve y=at the point(1,0).,Using ln x4=4 ln x,The quotient rule,When x=1,The gradient of the tangent at the point(1,0)is 4.,Us
32、e y y1=m(x x1)to find the equation of the tangent at the point(1,0).,y 0=4(x 1),y=4x 4,=4,Remember that ln 1=0,The chain ruleConnected rates of changeThe relationship betweenDifferentiating ex and related functionsDifferentiating ln x and related functionsThe product ruleThe quotient ruleExamination
33、-style questions,Contents,Boardworks Ltd 2006,46 of 53,Examination-style questions,Examination-style question 1,The height of a conical container is twice its radius,r cm,as shown in the diagram.,Liquid is poured into the container at a rate of 5 litres per minute.,If x cm is the depth of the liquid
34、 at time t minutes,write an expression for the rate at which the depth is increasing when x=2 cm.,(The volume of a cone of radius r and height h is given by).,The volume of the liquid in the container at time t is given by,Examination-style question 1,The liquid enters the container at a rate of 5 l
35、itres per minute so,Since the radius of the container is half its depth this can be written in term of x as,Examination-style question 1,So,Using the chain rule,At the instant when x=2 cm the rate at which the liquid is entering the container is,Examination-style question 2,a)Using the quotient rule
36、:,Examination-style question 2,b)When f(x)=0,When x=2,When x=2,Therefore,the graph of the function has turning points at(2,)and(2,).,Examination-style question 2,Looking at the gradient just before and just after x=2:,So(2,)is a maximum point.,Looking at the gradient just before and just after x=2:,So(2,)is a minimum point.,ive,0,+ive,0.01,0,0.01,0.01,0,0.01,ive,0,+ive,Examination-style question 2,When x=0,c)The curve crosses the axes when x=0 and when y=0.,y=0.,(Also,when y=0,x=0).,Therefore the curve has one crossing point at the origin,a minimum at(2,)and a maximum at(2,):,Also,0,and,0+.,
