Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin f ( ( cos f ( ( tan f ( ( tan [ f ( ] [ f ( ] [ f ( ]. f '(. f '(. f '( Stuents must recall the ifferentiation of basic functions (i.e. polynomial, eponential, logarithmic an trigonometric, which is assume to be covere uner Aitional Mathematics. Stuents must also be able to apply prouct an quotient rule to these basic functions, when require.. Implicit Differentiation Implicit ifferentiation involves ifferentiating a variable w.r.t. another variable. This technique is important in application problems involving equations of tangent an normal as well as rates of change. Note: Implicit ifferentiation is applie to epressions when or y cannot be mae as the subject. An eample is y y 0. 009 Mr Teo www.teachmejcmath-sg.webs.com
Special Eample on Implicit : Summary: Differentiation Stuents shoul be familiar with the following ifferentiation result: This result works for any real numbers of n. n n ( n When n takes a variable, such as or f(, this result cannot be applie. Stuents shoul use implicit ifferentiation to solve such a problem. For instance, Fin the erivative of, The solution is as shown, Let Taking i. e. ln y ln ln y ln Differentiate implicitly wrt, y y ln of both sies ln ( ln y ( ln I strongly avise stuents to know the restrictions of the formulas for all techniques of ifferentiation to avoi losing precious marks.. Differentiation involving Parametric Equations When fining for parametric equations, stuents must apply the chain rule i.e. 009 Mr Teo www.teachmejcmath-sg.webs.com
Note: Stuents nee not convert the parametric equations to its Cartesian form as chain rule offers the irect route to. If conversion to Cartesian form is require, stuents can apply substitution technique where applicable. Do be informe that some parametric epressions cannot be converte to its Cartesian form that easily. Applications of Differentiation These applications inclue i Equations of tangent an normal to a curve ii Rates of change iii Maima & minima iv Maclaurin s Series Note: Applications (i to (iii have been covere in etail uner Aitional Mathematics. The approach use to solve application problems in H Mathematics is the same, however, emphasis will be place on the ifferentiation techniques covere in H Mathematics (i.e. implicit an inverse trigonometry.. Equations of tangent an normal to a curve (For H an H stuents In solving these equations, stuents must recall the equation of the straight line i.e. y m c or y-y m( an that the relationship between their graients is m m -, where m an m are the graients of tangent an normal respectively. Stuents shoul epect the equation of the curve in the form f(,y constant, where f(,y contains an y variables in a single epression e.g. y y 0. Hence, the application of implicit ifferentiation is compulsory. For eample, let s etermine the graient for y y 0. 009 Mr Teo www.teachmejcmath-sg.webs.com
Applying implicit ifferentiation (an prouct rule, we will obtain Thus, Note: ( y y y 6 0 ( y y 6 ( y i In orer to efine the graients, we now nee the & y coorinates. Do note that if only one coorinate (e.g. is given, the other coorinate can be foun using the equation of the curve. If you recall, when you are solving A. Math problems, having -coorinate shoul suffice. ii For parametric equations, stuents must etermine the corresponing parametric values (e.g. t at the esire point. It is important for stuents to ientify that every point along the curve can be enote by (,y ((t, y(t. Also, if the sketch of the parametric equations is require, o sketch within the range of the parametric values provie. Special Note: It is very common to be aske for the tangents parallel to the & y aes. When the tangent is parallel to the -ais, 0 an the equation of line must be y a, where a is a constant. When tangent is parallel to the y-ais, tens to infinity or stuents can simply use 0. The equation of the line must then be b, where b is a constant.. Rates of Change (For H an H stuents Stuents must be familiar with the chain rule in orer to solve rates of change problems. It is also avisable that stuents apply implicit ifferentiation (i.e. ifferentiate w.r.t time. For eample, if we consier a rectangle with an y as the length an wih respectively, its area will be A y The rate of change of its area will then be, 009 Mr Teo www.teachmejcmath-sg.webs.com
A y, after applying implicit ifferentiation Note: Rates of change is always efine at an instance, either at a known length, epth or time. For more challenging problems, the instance is given inirectly. For eample, A container taking the shape of a cube is initially empty. Water is filling up at the rate of m /min. Fin the rate of change of its epth after minutes. Solution: Let be the sies of the cube. Hence, V V To fin the rate of change of its epth i.e., we nee the value of at the instant when t min. To fin the value of, we nee to etermine the volume after minutes i.e. 6m. V Thus, 6 ( 6. Maima & Minima (For H an H stuents The approach taken at H Mathematics is eactly similar to Aitional Mathematics. Stuents must perform the following steps: i Define area or volume of the require object (usually a proving question. ii Fin first erivative i.e. variable. A V or (usually involves ifferentiation with a single iii iv A V Solve for the stationary points i.e. 0 or 0. Define whether the area or volume is a ma or min using n erivative test (sometimes optional. 5 009 Mr Teo www.teachmejcmath-sg.webs.com
Note: The epressions for the area or volume are usually provie in the questions an stuents nee not worry if you cannot prove them i.e. Step (i. I highly recommen stuents to use the epression an continue with steps (ii to (iv to secure the remaining marks.. Maclaurin s Series Epansion (MSE (For H stuents only Special Note: Stuents shoul know by now that MSE is a compulsory question in H Mathematics (i.e. guarantee to be teste, other compulsory pure math sub-topics inclue Metho of Difference, Composite functions, MI, Loci for comple numbers, etc. Disclaimer: I may be wrong, so on t take my wor for it! The formula for MSE is provie in MF5, please refer. The stanar steps in efining the MSE are: i Prove an epression. (Stuents are then require to ifferentiate the epression repeately y ii Fin erivatives up to the require orer. (i.e. stop at if epansion to the is require iii Determine the values of each erivative at 0. iv Use the result to euce other results or to approimate values. Note: i For step (iv, the usual approaches to euce another MSE is to ( integrate ( ifferentiate ( apply substitution to the original result. For eample, From MF5, the MSE for ln( is To fin the MSE of ln(, stuents shoul observe that [ ln( ] 6 009 Mr Teo www.teachmejcmath-sg.webs.com
Hence, the MSE of Summary: Differentiation can be euce by ifferentiating all the terms in the MSE of ln( ln( i.e. [ ] Stuents may also use the following stanar series without proof. These are provie in MF5, please refer. Note: Stuents can apply substitution on these stanar series to obtain other MSE. Substitution is a very powerful an useful technique! For eample, To etermine the MSE of ln( sin, we can use the MSE of ln( an sin to euce. By comparing ln( sin with ln(, it is obvious that the substitution use is sin. ln( sin sin ( sin ( sin ( sin Since sin, we can now replace all the sin on the RHS accoringly.! 7 009 Mr Teo www.teachmejcmath-sg.webs.com
8 009 Mr Teo www.teachmejcmath-sg.webs.com!!!! sin ln( If terms with an above can be ignore, 6! sin ln( Note: Epan only the require terms.