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Basic Science, Engineering, New Cairo Campus Math I
2.2 Derivatives 1 . Definition of Derivative at a Point: The derivative of the function ?(?) at ? = ? is defined as ? ? = ??? ?→? ? ?+? −?(?) ? provided the limit exists. If the limit exists, we say that ? is differentiable at ? = ? , otherwise, we say that ? is not differentiable at ? = ? . An alternative form of ? ? = ??? ?→? ? ? −?(?) ?−? provided the limit exists. Remarks: ? ? = ൞ ? tangent line slope of tangent to ? = ? ? ?? (?? ? ) rate of change of?(?) at ? = ? velocity of the object at ? = ? when ? is a position functions 2
Example : Find the derivative of ? ? = ?? ? − ?? + ? at ? = ? . Carry out the same steps to find ? ??? : ? ? = ??? ?→? ? ?+? −?(?) ? : (1) ? ? = ?(?) ? − ? ? + ? = ? (2) ? ?+? −?(?) ? = ?(?+?) ? − ? ?+? + ?−? ? = ?+??+?? ? −?−??+?−? ? = ? + ?? ? ? = ? ?? + ? ? = ?? + ? (3) ? ? = ??? ?→? ? ?+? −?(?) ? = ??? ?→? ?? + ? = ? 3
Example: Find the derivative of ? ? = ?+? ??−? at ? = −? . (1) ? −? = ? −? = ? (2) ? −?+? −?(−?) ? = −?+?+? ?(−?+?)−? −? ? = ? ??−? ? = ? ?(??−?) = ? (??−?) (3) ? −? = ??? ?→? ? −?+? −?(−?) ? = ??? ?→? ? (??−?) = ? ? 4
Example : Find the derivative of ? ? = ?? − ? at ? = ? . (1) ? ? = ?(?) − ? = ? = ? (2) ? ?+? −?(?) ? = ?(?+?)−? − ? ? = ?+?? − ? ? = ? + ?? − ? ? ? + ?? + ? ? + ?? + ? = ? + ?? − ? ?( ? + ?? + ?) = ?? ?( ? + ?? + ?) = ? ( ? + ?? + ?) (3) ? ? = ??? ?→? ? ?+? −?(?) ? = ??? ?→? ? ( ?+??+ ?) = ? ( ?+ ?) = ? ? 5
2 . Definition of the Derivative Function: The derivative of the function ?(?) is the function ? ? is defined as ? ? = ??? ?→? ? ?+? −?(?) ? provided the limit exists. The process of computing a derivative is called differentiation. Example: Without computing, find the derivative function ?(?) for a . ? ? = ? b . ? ? = ?? + ? (? ? = ? , ? ? = ?. ? ? = ?? + ?, ? ? = ?) Example: Find ? ? if it exists where a . ? ? = ?? ? − ?? + ? ?. ? ? = ?+? ??−? ??? ? ≠ ?? ?. ? ? = ?? + ? ??? ? ≥ − ? ? 6
3. Sketching the Graph of ? ? from the Given Graph of ?(?) : From the definition of ? ? , we know ? ? = ? ??? at the point (𝐚, ?(?)) . So, for a given graph of ? we can sketch a rough graph of ? ? by estimate ? ??? at as many points as possible. Note: we should know exactly where ? ? = ? , ? ? > ? , ? ? < ? and where ? ? does not exist. Example: Page 166: 13-18, 27 7
4. Alternative Derivative Notations : The following are all alternative for the derivative function: let ? = ?(?) ?′(?) = ? = ?? ?? = ?? ?? = ? ?? ?(?) ? ?? is also called a differential operator . 8
5 . Nondifferentiable Points of a Function: From the definition of ? ? , we know the function is not differentiable at ? = ? if the limit ??? ?→? ? ?+? −?(?) ? does not exist. Conditions with which ? ? does not exist: (i) ? is not continuous at ? = ? ( ?(?) is not defined; ??? ?→? ? ? ?𝑵? or ??? ?→? ? ? ≠ ?(?) ; (ii) ??? ?→? ? ?+? −?(?) ? = ∞ ; or (iii) ??? ?→? ? ?+? −?(?) ? = ?𝑵? . ( ? is always continuous at ? = ? if ? ′(?) exists. ) Example: Page 167: 19-22. 9
Example : Let ? ? = ൝ ? ? ?? ? ≥ ? −? ?? ? < ? . Show graphically and algebraically that ? is continuous at ? = ? but ?(?) is not differentiable at ? = ?. Graphically, we see ? is continuous at ? = ? and is not differentiable at ? = ? Algebraically, ? is continuous at ? = ? because ??? ?→? + ? ? = ??? ?→? + ? ? = ?, ??? ?→? ? ? = ??? ?→? ? ? = ? = ?(?) Now check ??? ?→? ? ?+? −?(?) ? ??? ??? ?→? + ? ?+? −?(?) ? : 10
? ? + ? − ?(?) ? = ? ? − ? ? = ? ? ? = ? ?? ? > ? ? ? = −? ?? ? < ? ??? ?→? ? ? + ? − ?(?) ? = ??? ?→? −? = −? ??? ?→? + ? ? + ? − ?(?) ? = ??? ?→? + ? = ? Hence, ? is not differentiable at ? = ? since ??? ?→? ? ?+? −?(?) ? ?𝑵? 11
2.3 Computation of Derivatives: The Power Rule 1 . The Power Rule: Power function: ? ? = ? ? , ? a nonnegative integer Example : ?, ?, ? ? , . . . ? ???? . Graphically, we know the derivative of a power function exists everywhere. What is the derivative of a power function? Power Rule: For any nonnegative integer n , ? ? = ? ? , ?′ ? = ? ? ?−? Generalized Power Rule: For any real number r , ? ?? ? ? = ?? ?−?
Proof: ? ? = ??? ?→? ? ? + ? − ? ? ? = ??? ?→? ? + ? ? − ? ? ? = ??? ?→? ? ? + ?? ?−? ? + ⋯ + ??? ?−? + ? ? − ? ? ? = ??? ?→? ?? ?−? ? + ⋯ + ??? ?−? + ? ? ? = ??? ?→? ?(?? ?−? + ⋯ + ??? ?−? + ? ?−? ) ? = ??? ?→? (?? ?−? + ⋯ + ??? ?−? + ? ?−? ) = ?? ?−?
Example: Find ? ? ?? ?. ? ? = ? ?? , ??. ? ? = ? ???? ?. ?′ ? = ??? ?? , ??. ?′ ? = ???? ? ???? Example: Find ? ? if ?. ? ? = ? ? ? ? ? ??. ? ? = ? ? ? ? ?. ? ? = ? ? = ? −?/? , ? ? = ? ? ? −?/? ??. ? ? = ? ? ? = ? ?/? , ? ? = ? ? ? ?/?
2 . Differentiation Rule for ?? and ? ± ? : Let f and g be differentiable at x and c be a constant. Then 1. ? ?? ?(?) ± ?(?) = ?′(?) ± ?′(?) 2. ? ?? ? ?(?) = ? ?′(?) Example : Find ? ′(?) where i. ? ? = ?? ?? ? ? + ? ? ? ? + 𝝅 ii. ? ? = ? ? − ? ?? ? + ? − ? iii. ???. ? ? = ?? ? −? ? −? ?
Answer: i. ? ? = ?? ?? ? ? + ? ? ? ? + 𝝅 , ? ? = ??? ? + ? ? ? ? ? + ?? ? ? ?/? ii. ? ? = ?? ? − ?? ? + ? ? − ? − ? ? + ? = ?? ? − ? ? − ? ? − ? + ? , ? ? = ??? ? − ?? ? − ?? − ? iii. ? ? = ?? − ?? ? ? − ? −? , ?