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Derivation
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- 3. 3 “Differentiation is a process of finding a rate of change of one variable with respect to other.” • A change in variable is represented by: 𝑑𝑦 𝑑𝑥 “ ”
- 4. Differentiation is the part of the science of the Calculus, which was firstly developed by • ISSAC NEWTON • GOTTFRIED LEIBNIZ HISTORY 4 “in 17th Century”
- 5. WHY WE USE ? • • • • • Maximum and Minimum values Of particular Fucntions: 5
- 7. “SUM RULE: 7 In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y) + g(y))= 𝑑 𝑑𝑥 (f(y)) + 𝑑 𝑑𝑥 (g(y))
- 8. 8 EXAMPLE OF SUM RULE Find the Derivative of y=12x+36 using the sum rule. y=12x+36 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (12x + 36) 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 12x + 𝑑 𝑑𝑥 (36) 𝒅𝒚 𝒅𝒙 = 𝟏𝟐 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 12 . 1
- 9. “DIFFERENCE RULE: 9 In calculus, the Difference rule in differentiation is a method of finding the derivative of a function that is the difference of two other functions for which derivatives exist. The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y) − g(y))= 𝑑 𝑑𝑥 (f(y)) − 𝑑 𝑑𝑥 (g(y))
- 10. 10 EXAMPLE OF DIFFERENCE RULE Find the Derivative of y=-12x-104 using the Difference rule. y=34x-12x-104 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (-12x-104) 𝑑 𝑑𝑥 (y)=- 𝑑 𝑑𝑥 (12x) - 𝑑 𝑑𝑥 (104) 𝒅𝒚 𝒅𝒙 = -12
- 11. “POWER RULE: 11 “The power rule in calculus is a simple rule that helps you find the derivative of a variable raised to a power.” The power rule of an n-integer states: “ 𝑑 𝑑𝑥 𝑥 𝑛 =n. 𝑑 𝑑𝑥 𝑥 𝑛−1 ”
- 12. 12 EXAMPLE OF POWER RULE Find the derivative of y=𝑥5 -3𝑥2 +9x using the power rule. y=𝒙 𝟓 -3𝒙 𝟐 +9x Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (𝑥5 -3𝑥2 +9x) 𝑑 𝑑𝑥 (y)= 𝑑 𝑑𝑥 (𝑥5 )- 𝑑 𝑑𝑥 (3𝑥2 ) + 𝑑 𝑑𝑥 (9x) 𝒅𝒚 𝒅𝒙 = 𝟓𝒙 𝟒 − 𝟔x + 9
- 13. “PRODUCT RULE: 13 In Calculus, the Product rule is a formula used to find the derivatives of products of two or more function. It may be stated as: The sum rule states that for two functions f(y) and g(y): 𝑑 𝑑𝑥 (f(y).g(y))=f(y). 𝑑 𝑑𝑥 g(y) + g(y). 𝑑 𝑑𝑥 f(y)
- 14. 14 EXAMPLE OF PRODUCT RULE Find the derivative of y=𝑥2 𝑙𝑛𝑥 using the product rule. y=𝒙 𝟐 𝒍𝒏𝒙 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)=𝑥2 . 𝑑 𝑑𝑥 (lnx)+lnx 𝑑 𝑑𝑥 (𝑥2 ) 𝑑 𝑑𝑥 (y)=𝑥2 . 1 𝑥 + lnx.2x dy dx = x + 2xlnx 𝒅𝒚 𝒅𝒙 = 𝒙(𝟏 + 𝟐𝒍𝒏𝒙)
- 15. “QUOTIENT RULE: 15 “In Calculus, the Quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.” Let 𝑥 𝑦 be the function. 𝑑𝑦 𝑑𝑥 = 𝑑𝑥 𝑑𝑦 . 𝑦 − 𝑑𝑦 𝑑𝑥 . 𝑥 𝑦2 “ ”
- 16. 16 EXAMPLE OF QUOTIENT RULE Find the derivative of y= 𝟐𝒙+𝟓 𝟑𝒙−𝟐 using the quotient Rule..? y= 𝟐𝒙+𝟓 𝟑𝒙−𝟐 Differentiate w.r.t “x” into both sides 𝑑 𝑑𝑥 (y)= 2𝑥+5. 𝑑𝑦 𝑑𝑥 3𝑥−2 − 3𝑥−2 𝑑𝑦 𝑑𝑥 (2𝑥+5) (3𝑥−2)2 𝑑 𝑑𝑥 (y)= 2𝑥+5 . 3 − 3𝑥−2 .(2) (3𝑥−2)2
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