![Definition of the Derivative
• Notation used for the derivative given 𝑦 = 𝑓(𝑥):
• 𝑓′ 𝑥
• 𝑓′
• 𝑦′
•
𝑑𝑦
𝑑𝑥
• 𝐷𝑥[𝑓 𝑥 ]
• 𝐷𝑥(𝑦)](https://image.slidesharecdn.com/derivatives-240414225059-a4ab7eb4/85/DIFFERENTAL-CALCULUS-DERIVATIVES-FIRST-PART-7-320.jpg)
![Exercise
1.
𝑑
𝑑𝑥
𝑥
3
2
2.
𝑑
𝑑𝑥
(4
𝑥)
3.
𝑑
𝑑𝑥
(
3
𝑥7)
4. 𝐷𝑥(6)
5. 𝐷𝑥(9𝑥)
6. 𝐷𝑥
1
𝑥2
7.
𝑑𝑦
𝑑𝑥
if 𝑦 = 3𝑥2
8.
𝑑
𝑑𝑥
[ 4𝑥 2]
9. 𝐷𝑥(4𝑥2 + 10𝑥)
10. 𝐷𝑥(𝑥7 − 3 cos 𝑥 + cot 𝑥 − 𝑒𝑥 −
sech 𝑥 + csc−1
𝑥 + 8)
• Solve for the following derivatives](https://image.slidesharecdn.com/derivatives-240414225059-a4ab7eb4/85/DIFFERENTAL-CALCULUS-DERIVATIVES-FIRST-PART-12-320.jpg)
DIFFERENTAL CALCULUS DERIVATIVES FIRST PART
- 1. DERIVATIVES
- 2. Definition and Notation • Line 𝑙 is the tangent line of the graph of 𝒚 = 𝒇(𝒙) at point 𝑷. • It touches the graph of 𝑦 = 𝑓(𝑥) at point 𝑃(𝑥0, 𝑓 𝑥0 ) • Slope of the line 𝑙 • 𝑚𝑙 = lim Δ𝑥→0 𝑓 𝑥0+Δ𝑥 −𝑓 𝑥0 Δ𝑥 • Equation of line 𝑙 • If 𝑚𝑙 exists: 𝑦 − 𝑓 𝑥0 = 𝑚𝑙(𝑥 − 𝑥0) • If 𝑚𝑙 = 0: 𝑦 = 𝑓(𝑥0) • If 𝑚𝑙 approaches ±∞: 𝑥 = 𝑥0
- 3. Tangent Line • The slope of the tangent line at a certain point gives us an idea on the graph at that point. • Positive slope: graph is rising • Negative slope: graph is falling • Large slope: graph is relatively steep • Small slope: graph is relatively flat • Tangent line in this case does not necessarily touch the graph only once. • It touches the graph once, locally (at a small portion of the graph) • It may touch the graph again at other “far” points
- 4. Normal Line • Line 𝑘 is the normal line of the graph of 𝒚 = 𝒇(𝒙) at point 𝑷. • It touches the graph of 𝑦 = 𝑓(𝑥) at point 𝑃(𝑥0, 𝑓 𝑥0 ) • It is perpendicular to the tangent line • Slope of the line 𝑘 • 𝑚𝑘 = − 1 𝑚𝑙 • Equation of line 𝑘 • If 𝑚𝑘 exists: 𝑦 − 𝑓 𝑥0 = 𝑚𝑘(𝑥 − 𝑥0) • If 𝑚𝑘 = 0: 𝑦 = 𝑓(𝑥0) • If 𝑚𝑘 approaches ±∞: 𝑥 = 𝑥0
- 5. Definition of the Derivative • The derivative of a function 𝑓 𝑥 , denoted by 𝑓′(𝑥) is: 𝑓′ 𝑥 = lim Δ𝑥→0 𝑓 𝑥 + Δ𝑥 − 𝑓 𝑥 Δ𝑥 • 𝑓′(𝑥0) is the slope of the tangent line to the graph of 𝑓(𝑥) at point 𝑥0, 𝑓 𝑥0
- 6. Definition of the Derivative • The derivative of a function 𝑓 𝑥 at 𝑥 = 𝑥0, denoted by 𝑓′(𝑥0) is: 𝑓′ 𝑥0 = lim Δ𝑥→0 𝑓 𝑥0 + Δ𝑥 − 𝑓 𝑥0 Δ𝑥 • Or if we set Δ𝑥 = 𝑥 − 𝑥0: 𝑓′ 𝑥0 = lim (𝑥−𝑥0)→0 𝑓 𝑥0 + (𝑥 − 𝑥0) − 𝑓 𝑥0 (𝑥 − 𝑥0) 𝑓′ 𝑥0 = lim 𝑥→𝑥0 𝑓 𝑥 − 𝑓 𝑥0 𝑥 − 𝑥0 • The process of solving for the derivative is called differentiation.
- 7. Definition of the Derivative • Notation used for the derivative given 𝑦 = 𝑓(𝑥): • 𝑓′ 𝑥 • 𝑓′ • 𝑦′ • 𝑑𝑦 𝑑𝑥 • 𝐷𝑥[𝑓 𝑥 ] • 𝐷𝑥(𝑦)
- 8. Differentiation Rules • Let 𝑓, 𝑔, and ℎ be functions and 𝑐 is a real number. • If 𝑓 𝑥 = 𝑐 ∈ ℝ, then 𝑓′ 𝑥 = 0. • If 𝑓 𝑥 = 𝑐 ⋅ 𝑔 𝑥 , then 𝑓′ x = c ⋅ 𝑔′ 𝑥 if 𝑔′(𝑥) exists. • If ℎ 𝑥 = 𝑓 𝑥 ± 𝑔 𝑥 , then ℎ′ 𝑥 = 𝑓′ 𝑥 ± 𝑔′ 𝑥 ..
- 9. Differentiation Rules • Let 𝑓, 𝑔, and ℎ be functions and 𝑐 is a real number. • Power Rule: • If 𝑓 𝑥 = 𝑥𝑛, where 𝑛 is a rational number, then 𝑓′ 𝑥 = 𝑛𝑥𝑛−1. • Product Rule: • If ℎ 𝑥 = 𝑓 𝑥 𝑔(𝑥), then ℎ′ 𝑥 = 𝑓 𝑥 𝑔′ 𝑥 + 𝑔 𝑥 𝑓′ 𝑥 . • Quotient Rule • If ℎ 𝑥 = 𝑓 𝑥 𝑔 𝑥 where 𝑔 𝑥 ≠ 0, then ℎ′ 𝑥 = 𝑔 𝑥 𝑓′ 𝑥 −𝑓 𝑥 𝑔′ 𝑥 𝑔 𝑥 2 .
- 10. Differentiation Rules • The following differentiation formulas hold: 1. d dx sin 𝑥 = cos 𝑥 2. 𝑑 𝑑𝑥 cos 𝑥 = − sin 𝑥 3. 𝑑 𝑑𝑥 tan 𝑥 = sec2 𝑥 4. 𝑑 𝑑𝑥 cot 𝑥 = − csc2 𝑥 5. 𝑑 𝑑𝑥 sec 𝑥 = sec 𝑥 tan 𝑥 6. 𝑑 𝑑𝑥 csc 𝑥 = − csc 𝑥 cot 𝑥 7. 𝑑 𝑑𝑥 ln 𝑥 = 1 𝑥 8. 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑥 9. 𝑑 𝑑𝑥 𝑎𝑥 = 𝑎𝑥 ln 𝑎 , 𝑎 > 0, 𝑎 ≠ 1 10. 𝑑 𝑑𝑥 log𝑎 𝑥 = 1 ln 𝑎 𝑥 , 𝑎 > 0, 𝑎 ≠ 1
- 11. Differentiation Rules • The following differentiation formulas hold: 11. 𝑑 𝑑𝑥 sin−1 𝑥 = 1 1−𝑥2 12. 𝑑 𝑑𝑥 cos−1 𝑥 = − 1 1−𝑥2 13. 𝑑 𝑑𝑥 tan−1 𝑥 = 1 1+𝑥2 14. 𝑑 𝑑𝑥 cot−1 𝑥 = − 1 1+𝑥2 15. 𝑑 𝑑𝑥 sec−1 𝑥 = 1 𝑥 𝑥2−1 16. 𝑑 𝑑𝑥 csc−1 𝑥 = − 1 𝑥 𝑥2−1 17. 𝑑 𝑑𝑥 sinh 𝑥 = cosh 𝑥 18. 𝑑 𝑑𝑥 cosh 𝑥 = sinh 𝑥 19. 𝑑 𝑑𝑥 tanh 𝑥 = sech2 𝑥 20. 𝑑 𝑑𝑥 coth 𝑥 = − csch2 𝑥 21. 𝑑 𝑑𝑥 sech 𝑥 = − sech 𝑥 tanh 𝑥 22. 𝑑 𝑑𝑥 csch 𝑥 = − csch 𝑥 coth 𝑥
- 12. Exercise 1. 𝑑 𝑑𝑥 𝑥 3 2 2. 𝑑 𝑑𝑥 (4 𝑥) 3. 𝑑 𝑑𝑥 ( 3 𝑥7) 4. 𝐷𝑥(6) 5. 𝐷𝑥(9𝑥) 6. 𝐷𝑥 1 𝑥2 7. 𝑑𝑦 𝑑𝑥 if 𝑦 = 3𝑥2 8. 𝑑 𝑑𝑥 [ 4𝑥 2] 9. 𝐷𝑥(4𝑥2 + 10𝑥) 10. 𝐷𝑥(𝑥7 − 3 cos 𝑥 + cot 𝑥 − 𝑒𝑥 − sech 𝑥 + csc−1 𝑥 + 8) • Solve for the following derivatives
- 13. Exercise Solve for the following derivatives 11. 𝑑 𝑑𝑥 𝑥 − 3 2𝑥2 − 3 12. 𝐷𝑥 3𝑥5 + 6𝑥 4 3 − 2 4𝑥3 − 53 𝑥 13. 𝑑𝑦 𝑑𝑥 if 𝑦 = 2𝑥3− 1 𝑥3+4 3𝑥−5 14. 𝑓′(𝑥) if 𝑓 𝑥 = 2𝑥4− 𝑥+2 2𝑥3−4 𝑥−3𝑥 15. 𝐷𝑥 𝑥3 − 1 𝑥3 16. 𝑑 𝑑𝑥 sec 𝑥 csc 𝑥 17. 𝐷𝑥 cot 𝑥−𝑥 1+tan 𝑥 18. 𝑑 𝑑𝑥 (sin 2𝑥) 19. 𝐷𝑥 2𝑥 cos 𝑥 20. 𝑑 𝑑𝑥 𝑥−8 4 𝑥 sec 𝑥−3 cot 𝑥
- 14. Answers to Exercises 1. 3 2 𝑥 1 2 2. 1 4 4 𝑥3 3. 7 3 3 𝑥4 4. 0 5. 9 6. − 2 𝑥3 7. 6𝑥 8. 32𝑥 9. 8𝑥 + 10 10. 7𝑥6 + 3 sin 𝑥 − csc2 𝑥 − 𝑒𝑥 + sech 𝑥 tanh 𝑥 − 1 𝑥 𝑥2−1
- 15. Answers to Exercises 11. 6𝑥2 − 12𝑥 − 3 12. 3𝑥5 + 6𝑥 4 3 − 2 12𝑥2 − 5 3 3 𝑥2 + (4𝑥3 − 16. sec 𝑥 tan 𝑥 csc 𝑥 + (sec 𝑥)(− csc 𝑥 cot 𝑥) 17. 1+tan 𝑥 − csc2 𝑥−1 − cot 𝑥−𝑥 sec2 𝑥 1+tan 𝑥 2 18. 2 sin 𝑥 − sin 𝑥 + cos 𝑥 cos 𝑥 19. 2𝑥 − sin 𝑥 + (cos 𝑥)(2) 20. sec 𝑥−3 cot 𝑥 1− 2 4 𝑥3 − 𝑥−84 𝑥 sec 𝑥 tan 𝑥+3 csc2 𝑥 sec 𝑥−3 cot 𝑥