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23&24. Derivatives of exponential logarithmic and trigonometric functions.

23&24. Derivatives of exponential logarithmic and trigonometric functions. A.SERRA. 23&24. Very important note:. Please note that units 23 and 24 are related to the following units: 3 (Exponents), 4 (Logarithms) 5 (Natural Logarithms). 10&11 (Trigonometry)

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23&24. Derivatives of exponential logarithmic and trigonometric functions.

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  1. 23&24. Derivatives of exponential logarithmic and trigonometric functions. A.SERRA

  2. 23&24. Very important note: • Please note that units 23 and 24 are related to the following units: • 3 (Exponents), • 4 (Logarithms) • 5 (Natural Logarithms). • 10&11 (Trigonometry) • YOU need to revise these units to be able to follow 23&24!

  3. 23A. Derivatives of exponential functions • Class work: Geogebra Investigation • First Conclusion: The derivative of exis quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex! • What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph. • If u is a function of x, we can obtain the derivative of an expression in the form eu: • Chain rule! • Second conclusion: If we have an exponential function with some base b, we have the following derivative: • Please note that b^u = e^lnb^u (see 23b) • INDIVIDUAL WORK: • Do and check examples 1,2,3 to make sure you understand this. • Exercise 23A:1 (do half of them only

  4. 23B. Using natural logarithms. • Please note thatthis section isrevisionfrom last year. • Wewill not becoveringthis in class. • INDIVIDUAL WORK • Do and check examples 4, 5, 6 and 7 to make sure you can use logarithms • Exercises 23B: 3,5.

  5. 23C. Derivatives of logarithmic functions • Group work: • GeoGebra investigation: derivatives of logarithmicfunctions. • d (ln(x))/dx = 1/x • d (logb(x))/dx = 1/x ln(b) • INDIVIDUAL WORK • Do and check examples 8 and 9 to make sure you understand this. • Exercise 23C: 1 (a,d,g,j,m), 3(j,k,l), 5

  6. 23D. Applications • Please note: these are exam-type exercises. • Group work: • Studentschoose an exercisefrom the extra list. • Teacherhelpsstudentswork out the solution step by step. • INDIVIDUAL WORK • Do and check example 10 if you need to. • Exercise 23D: 1, 5, 9, 13, 17, 21

  7. 24A. Derivative of sin x, cos x, tan x • Group work: • Geogebraexperiment: derivatives of trigonometricfunctions. • d (sin(x))/dx = cos(x) • d (cos(x))/dx = -sin(x) • d (tan(x))/dx = 1/(cos(x))^2 • NB: x= angle in radians. • INDIVIDUAL WORK • Do and check examples 1,2,3 and 4 • Exercise 24A: 1 (a,d,g), 3 (a,e,i), 9, 11

  8. 24B. Optimization with trigonometry. • Group work: • Exercise 24B 2, 4 or 6 • INDIVIDUAL WORK • Exercise 24A: 1 (a,d,g), 3 (a,e,i), 9, 11 • Exercise 24 B 1,3 • Extra: any other exercise in 24B

  9. Homework: Review units 23 and 24. • INDIVIDUAL WORKHOMEWORK • Review Set 23A. (Do, correct your answers and write down score (total and percentage “%”) • Review Set 24. (Do, correct your answers and write down score (total and percentage “%”). • Extra: Review Set 23B • Mock Test Units 22,23,24 • Test Units 22,23,24

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